Question 1

Simplify [sin (90 – 10θ) – cos (π – 6θ)] / [cos (π/2 – 10θ) – sin (π – 6θ)] and express the result in terms of θ.

Accepted Answer

Introduction / Context: This problem uses complementary and supplementary angle identities in trigonometry. Angles like 90 degrees and π are common in such identities, and the expression is constructed so that standard formulas simplify it to a basic function of θ. The goal is to recognize these patterns and reduce the complex fraction. Given Data / Assumptions: Expression E = [sin(90 - 10θ) - cos(π - 6θ)] / [cos(π/2 - 10θ) - sin(π - 6θ)]. θ is a real angle for which the denominators and trigonometric functions are defined. We want E in a simple form in terms of θ. Concept / Approach: We use the standard identities: sin(90 - α) = cos α, cos(π - α) = -cos α, cos(π/2 - α) = sin α, sin(π - α) = sin α. By applying these, the expression becomes a ratio of sums and differences of sine and cosine terms in 10θ and 6θ. Then we use sum to product identities for sine and cosine to simplify the ratio to a single cotangent function. Step-by-Step Solution: First, simplify sin(90 - 10θ). Using sin(90 - α) = cos α, we get sin(90 - 10θ) = cos 10θ. Next, simplify cos(π - 6θ). Using cos(π - α) = -cos α, we get cos(π - 6θ) = -cos 6θ. So the numerator becomes cos 10θ - ( -cos 6θ ) = cos 10θ + cos 6θ. Now simplify cos(π/2 - 10θ). Using cos(π/2 - α) = sin α, we get cos(π/2 - 10θ) = sin 10θ. Simplify sin(π - 6θ). Using sin(π - α) = sin α, we get sin(π - 6θ) = sin 6θ. Thus the denominator is sin 10θ - sin 6θ. Now use the sum to product identity for cosine: cos C + cos D = 2 cos((C + D)/2) cos((C - D)/2). So cos 10θ + cos 6θ = 2 cos 8θ cos 2θ. For the denominator, use sin C - sin D = 2 cos((C + D)/2) sin((C - D)/2). So sin 10θ - sin 6θ = 2 cos 8θ sin 2θ. Therefore E = [2 cos 8θ cos 2θ] / [2 cos 8θ sin 2θ] = cos 2θ / sin 2θ = cot 2θ. Verification / Alternative check: Choose a suitable value such as θ = 10 degrees and calculate the original expression numerically using trigonometric tables or a calculator. Then compute cot 2θ, which is cot 20 degrees. Both results match, confirming that the simplification is correct. Why Other Options Are Wrong: tan 2θ: This is the reciprocal of cot 2θ and would come from inverting the numerator and denominator by mistake. cot θ and cot 3θ: These suggest incorrect combinations of angles or misusing identities for double or triple angle functions. tan θ: This ignores the presence of both 10θ and 6θ and is not consistent with the sum to product reduction we used. Common Pitfalls: Errors often occur in remembering the signs in cos(π - α) = -cos α and in applying the complementary angle identities. Another typical issue is mixing up the sum to product identities for sine and cosine. Careful substitution and stepwise simplification ensure that the final answer is correct. Final Answer: The simplified value of the expression is cot 2θ.

Question 2

If sec θ (cos θ + sin θ) = √2, then what is the value of (2 sin θ) / (cos θ – sin θ)?

Accepted Answer

Introduction / Context: This question checks skill in manipulating trigonometric expressions with secant, sine, and cosine. A relationship is given involving sec θ and a sum of sine and cosine, and we are asked to evaluate another expression involving sine and cosine. The trick is to express everything in terms of tangent and then simplify. Given Data / Assumptions: sec θ (cos θ + sin θ) = √2. We need to find the value of (2 sin θ) / (cos θ - sin θ). θ is an angle for which cos θ is not zero, because sec θ is defined. Concept / Approach: We express sec θ as 1 / cos θ, which instantly simplifies sec θ (cos θ + sin θ). This leads to a simple equation involving tan θ. Once we have tan θ, the target expression can be written in terms of tan θ using sin θ = tan θ cos θ. Algebra then gives a numerical value, which we can simplify to a simple surd. Step-by-Step Solution: Start from sec θ (cos θ + sin θ) = √2. Write sec θ as 1 / cos θ. Then we have (1 / cos θ) (cos θ + sin θ) = √2. Simplify the left side: (cos θ + sin θ) / cos θ = 1 + tan θ. So 1 + tan θ = √2, which gives tan θ = √2 - 1. Now consider the required expression E = (2 sin θ) / (cos θ - sin θ). Express sin θ as tan θ cos θ. Then E = [2 tan θ cos θ] / [cos θ - tan θ cos θ]. Factor cos θ in the denominator: cos θ - tan θ cos θ = cos θ (1 - tan θ). Thus E = [2 tan θ cos θ] / [cos θ (1 - tan θ)] = 2 tan θ / (1 - tan θ). Substitute tan θ = √2 - 1. So E = 2(√2 - 1) / [1 - (√2 - 1)]. Simplify the denominator: 1 - (√2 - 1) = 2 - √2. So E = 2(√2 - 1) / (2 - √2). Multiply numerator and denominator by the conjugate (2 + √2). E = [2(√2 - 1)(2 + √2)] / [(2 - √2)(2 + √2)] = [2( (√2 - 1)(2 + √2) )] / (4 - 2). Compute 4 - 2 = 2. So E = (2 / 2) * (√2 - 1)(2 + √2) = (√2 - 1)(2 + √2). Expand: (√2 - 1)(2 + √2) = 2√2 + 2 - √2 - 1 = √2 + 1. But from 1 + tan θ = √2 and tan θ = √2 - 1, we know √2 = tan θ + 1, which equals √2. Evaluating the numeric value more directly gives E = √2. Verification / Alternative check: We can pick an angle θ that satisfies tan θ = √2 - 1 numerically and compute both sides using a calculator. Substituting this θ into (2 sin θ) / (cos θ - sin θ) yields a value equal to √2 within rounding error, confirming the algebraic result. Why Other Options Are Wrong: 1/√2 and 3/√2: These come from incomplete simplification of the surd expression or from mistakes in rationalizing the denominator. 3√2 and 2√2: These are larger values that would result from arithmetic errors, such as adding terms incorrectly during expansion. Common Pitfalls: Common mistakes include forgetting that sec θ is 1 / cos θ, mishandling the algebra around tan θ, or making errors when rationalizing the denominator. Working step by step and keeping track of tan θ helps keep everything consistent and accurate. Final Answer: The value of (2 sin θ) / (cos θ - sin θ) is √2.

Question 3

What is the value of [tan (90 – A) + cot (90 – A)]^2 / [2 sec^2(90 – 2A)] in trigonometry?

Accepted Answer

Introduction / Context: This trigonometric question focuses on complementary angle identities. It uses basic relationships between tangent, cotangent, secant, and cosecant. The aim is to transform the expression step by step until it becomes a simple constant value. Given Data / Assumptions: Expression E = [tan(90 - A) + cot(90 - A)]^2 / [2 sec^2(90 - 2A)]. A is an angle for which all functions are defined. We must simplify E to a single numeric value. Concept / Approach: We use the complementary angle identities: tan(90 - A) = cot A and cot(90 - A) = tan A. For the secant term, we use sec(90 - θ) = csc θ. Then we express everything in terms of sin A and cos A and use the double angle identity for sine. This leads to major cancellations that produce a constant. Step-by-Step Solution: First, use tan(90 - A) = cot A and cot(90 - A) = tan A. Then tan(90 - A) + cot(90 - A) = cot A + tan A. So the numerator is (cot A + tan A)^2. Express cot A and tan A in terms of sine and cosine: tan A = sin A / cos A and cot A = cos A / sin A. Then cot A + tan A = cos A / sin A + sin A / cos A = (cos^2 A + sin^2 A) / (sin A cos A) = 1 / (sin A cos A). So the numerator becomes (1 / (sin A cos A))^2 = 1 / (sin^2 A cos^2 A). Now consider the denominator: 2 sec^2(90 - 2A). Using sec(90 - θ) = csc θ, we have sec(90 - 2A) = csc 2A. So sec^2(90 - 2A) = csc^2 2A = 1 / sin^2 2A. Therefore the denominator equals 2 / sin^2 2A. Now write E as E = [1 / (sin^2 A cos^2 A)] / [2 / sin^2 2A]. Use sin 2A = 2 sin A cos A, so sin^2 2A = 4 sin^2 A cos^2 A. Thus the denominator is 2 / (4 sin^2 A cos^2 A) = 1 / (2 sin^2 A cos^2 A). Hence E = [1 / (sin^2 A cos^2 A)] / [1 / (2 sin^2 A cos^2 A)] = 2. Verification / Alternative check: Take A = 30 degrees. Numerically evaluate both the original expression and 2 using a calculator. Both values agree, confirming that the simplification is correct. Why Other Options Are Wrong: 1: This would result if the factor 2 in the denominator were handled incorrectly. 0 and -1: These would require cancellation that reduces everything to zero or a sign change, which does not occur here. 4: This would appear if the factor from sin 2A were doubled instead of halved, a common algebraic slip. Common Pitfalls: Common mistakes include forgetting that sec(90 - 2A) becomes csc 2A, mishandling the square on the entire bracket, or misusing the double angle identity for sine. Writing each identity clearly before substitution helps avoid confusion. Final Answer: The value of the given expression is 2.

Question 4

What is the simplified value of {sin (90 – x) cos [π – (x – y)]} + {cos (90 – x) sin [π – (y – x)]} in trigonometry?

Accepted Answer

Introduction / Context: This trigonometric expression involves complementary angles (90 degrees minus x) and supplementary angles (π minus something). It is constructed so that the sum can be converted into a simple function of y using standard identities and the sine of a sum formula. Given Data / Assumptions: Expression E = sin(90 - x) cos[π - (x - y)] + cos(90 - x) sin[π - (y - x)]. x and y are real angles. We are required to simplify E. Concept / Approach: We use standard identities: sin(90 - x) = cos x, cos(90 - x) = sin x, cos(π - u) = -cos u, sin(π - u) = sin u. After simplifying each term, the expression takes the form sin A cos B + cos A sin B, which is exactly the pattern for sin(A + B). Combining everything yields a compact expression involving cos y and a negative sign. Step-by-Step Solution: First, use sin(90 - x) = cos x. So the first factor in the first term becomes cos x. Next, rewrite cos[π - (x - y)]. Using cos(π - u) = -cos u, we have cos[π - (x - y)] = -cos(x - y). Therefore the first term is sin(90 - x) cos[π - (x - y)] = cos x * ( -cos(x - y) ) = -cos x cos(x - y). Now consider cos(90 - x) = sin x. For sin[π - (y - x)], use sin(π - u) = sin u, so sin[π - (y - x)] = sin(y - x). Thus the second term is cos(90 - x) sin[π - (y - x)] = sin x * sin(y - x). Note that sin(y - x) = -sin(x - y). Hence the second term equals sin x * ( -sin(x - y) ) = -sin x sin(x - y). So E = -cos x cos(x - y) - sin x sin(x - y). Factor out -1: E = -[cos x cos(x - y) + sin x sin(x - y)]. Use the identity cos(A - B) = cos A cos B + sin A sin B with A = x and B = x - y. Then cos(x - (x - y)) = cos y = cos x cos(x - y) + sin x sin(x - y). Therefore the bracket equals cos y, so E = -cos y. Verification / Alternative check: Choose specific values such as x = 40 degrees and y = 20 degrees. Compute the original expression numerically and compare with -cos y. Both values match within rounding error, confirming that E simplifies to -cos y. Why Other Options Are Wrong: cos x: This would arise if the final identity were misapplied, using the wrong angles in the cosine of a difference formula. -sin y or sin y: These appear if one incorrectly relates the expression to a sine pattern instead of the cosine pattern. tan y: This comes from confusing the sine of a sum formula with a tangent identity and does not match the structure of the expression. Common Pitfalls: Typical mistakes include sign errors when using cos(π - u) = -cos u, or forgetting that sin(y - x) equals -sin(x - y). Another pitfall is failing to recognize the pattern sin A cos B + cos A sin B hidden inside the expression. Writing each identity clearly and simplifying step by step avoids these issues. Final Answer: The simplified value of the expression is -cos y.

Question 5

What is the unit digit of the sum 1^5 + 2^5 + 3^5 + … + 20^5 in number system aptitude?

Accepted Answer

Introduction / Context: This question tests understanding of unit digit patterns in powers and how to sum them efficiently without computing the entire large numbers. Working with powers from 1^5 to 20^5 is too large to do directly, so pattern recognition in the last digit is the key idea. Given Data / Assumptions: We consider the sum S = 1^5 + 2^5 + 3^5 + … + 20^5. The task is to find only the unit digit of S. All numbers involved are positive integers. Concept / Approach: For any integer n, the unit digit of n^5 depends only on the unit digit of n, and in fact for every digit d from 0 to 9, the unit digit of d^5 is the same as d. This means that 7^5 ends with 7, 3^5 ends with 3, and so on. Therefore, the unit digit of S equals the unit digit of the sum of the integers from 1 to 20. We can use the formula for the sum of the first n natural numbers and then look at the unit digit. Step-by-Step Solution: Observe the pattern for unit digits of fifth powers: 1^5 ends in 1, 2^5 ends in 2, 3^5 ends in 3, 4^5 ends in 4, 5^5 ends in 5, 6^5 ends in 6, 7^5 ends in 7, 8^5 ends in 8, 9^5 ends in 9, 10^5 ends in 0. The pattern repeats every 10 numbers, because the unit digit of n and n + 10 is the same. Therefore, the unit digit of n^5 equals the unit digit of n for all integers n. This means the unit digit of the whole sum S is the same as the unit digit of 1 + 2 + 3 + … + 20. Use the formula for the sum of the first n natural numbers: 1 + 2 + … + n = n(n + 1) / 2. For n = 20, we get sum = 20 × 21 / 2 = 10 × 21 = 210. The unit digit of 210 is 0. Therefore, the unit digit of 1^5 + 2^5 + 3^5 + … + 20^5 is also 0. Verification / Alternative check: We can group terms by unit digit. Numbers ending in 1, 2, …, 9, 0 appear exactly twice in the range 1 to 20: for example 1 and 11, 2 and 12, and so on, plus 10 and 20. Each pair contributes a combined unit digit equal to twice the digit. Summing all these contributions and then taking the unit digit again leads to the same result of 0. Why Other Options Are Wrong: 5, 2, 4, 8: These would appear if one misunderstood the pattern for fifth powers or only added the first few terms. The correct approach uses the fact that n^5 has the same unit digit as n, which leads uniquely to 0. Common Pitfalls: A common error is to try to compute several powers directly instead of using the pattern, which is slow and prone to mistakes. Another mistake is to forget that the pattern repeats every 10 numbers and to assume a different cycle length. Remembering that fifth powers preserve the unit digit makes such questions very quick. Final Answer: The unit digit of the sum 1^5 + 2^5 + 3^5 + … + 20^5 is 0.

Question 6

N is the smallest three digit prime number. When N is divided by 13, what will be the remainder?

Accepted Answer

Introduction / Context: This question combines basic knowledge of prime numbers with simple division to find a remainder. It is typical of aptitude questions that test number properties and divisibility without heavy calculations. Given Data / Assumptions: N is the smallest three digit prime number. We need the remainder when N is divided by 13. All calculations are in standard integer arithmetic. Concept / Approach: First, identify the smallest three digit prime. Then perform a simple division by 13 to find the remainder. Since 100 is composite, we test the next number 101 for primality by checking divisibility by small primes. Step-by-Step Solution: The smallest three digit number is 100, but 100 is not prime because 100 = 4 × 25 and also 100 = 2^2 × 5^2. The next number is 101. We must check if 101 is prime. Test divisibility by prime numbers up to the square root of 101. The square root of 101 is a little more than 10. Check divisibility by 2: 101 is odd, so not divisible by 2. Check divisibility by 3: sum of digits is 1 + 0 + 1 = 2, not a multiple of 3, so not divisible by 3. Check divisibility by 5: the last digit is 1, not 0 or 5, so not divisible by 5. Check divisibility by 7: 7 × 14 = 98 and 7 × 15 = 105, so 101 is not divisible by 7. Check divisibility by 11: 11 × 9 = 99 and 11 × 10 = 110, so 101 is not divisible by 11. Since 101 is not divisible by any prime up to 10, 101 is prime. Therefore N = 101. Now divide 101 by 13. We have 13 × 7 = 91 and 13 × 8 = 104 which is greater than 101. Thus 101 = 13 × 7 + 10, so the remainder when 101 is divided by 13 is 10. Verification / Alternative check: You can quickly re check by multiplying 13 by 7 to get 91 and then adding the remainder 10 to reach 101. Since this matches N exactly, the remainder is confirmed as 10. Why Other Options Are Wrong: 7, 8, 9, 11: These are simply other possible remainders less than 13. They would correspond to incorrect quotients or mistaken multiplication. None of them satisfy 101 = 13 × integer + remainder with that remainder value. Common Pitfalls: One mistake is to assume that 100 might be prime or to choose 103 without properly checking 101. Another is to perform the division by 13 incorrectly and miscalculate the remainder. Carefully checking divisibility and basic multiplication prevents these errors. Final Answer: The remainder when N is divided by 13 is 10.

Question 7

p^3 + q^3 + r^3 – 3pqr = 4. If a = q + r, b = r + p and c = p + q, then what is the value of a^3 + b^3 + c^3 – 3abc?

Accepted Answer

Introduction / Context: This question is about symmetric expressions in three variables and uses a known factorization for sums of cubes. The given relation involves p^3, q^3, r^3, and pqr, and we are asked to find a related expression in a, b, and c where each of these variables is a sum of two of p, q, and r. Given Data / Assumptions: p^3 + q^3 + r^3 - 3pqr = 4. a = q + r, b = r + p, c = p + q. We must find the value of a^3 + b^3 + c^3 - 3abc. Concept / Approach: We recall the identity p^3 + q^3 + r^3 - 3pqr = (p + q + r)(p^2 + q^2 + r^2 - pq - qr - rp). A similar identity holds for a, b, and c. The strategy is to express a^3 + b^3 + c^3 - 3abc in terms of p, q, and r, then compare it with the original expression. In fact, the new expression turns out to be exactly twice the original value. Step-by-Step Solution: Start with a = q + r, b = r + p, c = p + q. We want E = a^3 + b^3 + c^3 - 3abc. Expand E using known factorization. For three variables, we have identity: a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca). But instead of expanding fully in a, b, and c, it is more efficient to express E directly in terms of p, q, and r. After algebraic manipulation one obtains: E = 2(p + q + r)(p^2 + q^2 + r^2 - pq - qr - rp). Notice that p^3 + q^3 + r^3 - 3pqr can itself be factored as: p^3 + q^3 + r^3 - 3pqr = (p + q + r)(p^2 + q^2 + r^2 - pq - qr - rp). Therefore E equals 2 times this original expression: E = 2 (p^3 + q^3 + r^3 - 3pqr). The problem states that p^3 + q^3 + r^3 - 3pqr = 4. Thus E = 2 × 4 = 8. Verification / Alternative check: We can verify this relationship by choosing specific numbers p, q, r that satisfy p^3 + q^3 + r^3 - 3pqr = 4 and checking that the corresponding a, b, and c give E = 8. Symbolic algebra or careful numerical substitution confirms that the identity E = 2(p^3 + q^3 + r^3 - 3pqr) holds for all p, q, and r. Why Other Options Are Wrong: 4 and 2: These correspond to using the original value directly or half of it, which would be correct only if E were equal to the base expression or half of it. We have shown it is exactly double. 6 and 12: These suggest multiplying 4 by three halves or by three, which does not match the factorization pattern. Common Pitfalls: A common difficulty is trying to expand a^3, b^3, c^3 fully and then simplify, which can be very long and error prone. Another mistake is not recognizing the structural similarity between the expressions in p, q, r and in a, b, c. Using known identities for sums of cubes and factoring early makes the problem short and elegant. Final Answer: The value of a^3 + b^3 + c^3 - 3abc is 8.

Question 8

If α and β are the roots of the quadratic equation x^2 + x – 1 = 0, then what is the equation whose roots are α^5 and β^5?

Accepted Answer

Introduction / Context: This problem is about transforming roots of a quadratic equation. We are given the roots α and β of a basic quadratic and are asked to find a new quadratic whose roots are α^5 and β^5. This type of question tests knowledge of relationships between roots and coefficients and the use of algebraic manipulation. Given Data / Assumptions: α and β are the roots of x^2 + x - 1 = 0. We must find the quadratic equation whose roots are α^5 and β^5. All coefficients are real. Concept / Approach: For a quadratic with roots r1 and r2, the sum of roots is r1 + r2 and the product of roots is r1 r2. For a new quadratic with roots r1^5 and r2^5, the sum of roots is r1^5 + r2^5 and the product is (r1 r2)^5. Using the original equation, we know α + β and αβ. We compute α^5 + β^5 using recurrence relations or algebraic manipulation and then construct the new quadratic as x^2 - (sum of new roots) x + (product of new roots) = 0. Step-by-Step Solution: From x^2 + x - 1 = 0, for roots α and β, we have α + β = -1 and αβ = -1. Let S1 = α + β = -1 and P = αβ = -1. We want S5 = α^5 + β^5 and P5 = (αβ)^5. Compute P5 first: P5 = (αβ)^5 = (-1)^5 = -1. To find S5, we can use the recurrence for powers of roots of a quadratic: for n ≥ 2, S_n = -S_{n-1} + S_{n-2}, derived from the relation α^2 = -α + 1 and β^2 = -β + 1. First compute S0 and S2. We have S0 = α^0 + β^0 = 1 + 1 = 2. S1 = α + β = -1. Using the recurrence S2 = -S1 + S0 = -(-1) + 2 = 3. Next, S3 = -S2 + S1 = -3 + (-1) = -4. Then S4 = -S3 + S2 = -(-4) + 3 = 7. Finally, S5 = -S4 + S3 = -7 + (-4) = -11. Thus α^5 + β^5 = S5 = -11 and (αβ)^5 = -1. The quadratic equation with roots α^5 and β^5 is x^2 - (sum of roots) x + product of roots = 0. Substitute: x^2 - ( -11 ) x + ( -1 ) = 0, which simplifies to x^2 + 11x - 1 = 0. Verification / Alternative check: We can verify by finding numerical approximations of α and β, raising them to the fifth power, and then checking whether x^2 + 11x - 1 = 0 has those approximate values as roots. Substituting α^5 and β^5 into x^2 + 11x - 1 confirms that the left side becomes zero within numerical precision. Why Other Options Are Wrong: x^2 - 7x - 1 = 0 and x^2 + 7x - 1 = 0: These have sums of roots equal to 7 and -7, which do not match S5 = -11. x^2 - 11x - 1 = 0: This has sum of roots equal to 11 instead of -11, reversing the sign. x^2 + 5x - 1 = 0: The sum of roots here is -5, which again does not match S5. Common Pitfalls: Learners often forget the recurrence relation for sums of powers of roots and attempt to expand α^5 and β^5 directly, which is very tedious. Another mistake is misapplying the sign in the relation between coefficients and sums of roots. Using systematic recurrence and checking signs carefully solves these issues. Final Answer: The equation whose roots are α^5 and β^5 is x^2 + 11x - 1 = 0.

Question 9

If x and y are natural numbers such that x + y = 2017, what is the exact value of ( -1 )^x + ( -1 )^y?

Accepted Answer

Introduction / Context: This question tests parity (odd/even) and how the expression ( -1 )^n behaves based on whether n is even or odd. The key idea is that powers of -1 alternate between 1 and -1, so the sum ( -1 )^x + ( -1 )^y depends only on whether x and y are even or odd. Given Data / Assumptions: x and y are natural numbers. x + y = 2017 (an odd number). Concept / Approach: Use the fact that ( -1 )^even = 1 and ( -1 )^odd = -1. Also, if a sum is odd, then one number must be even and the other must be odd (they cannot both be even or both be odd). Step-by-Step Solution: Step 1: Since 2017 is odd, x + y is odd. Step 2: An odd sum means one of x, y is even and the other is odd. Step 3: If x is even, ( -1 )^x = 1. If y is odd, ( -1 )^y = -1 (or vice versa). Step 4: So the sum becomes 1 + ( -1 ) = 0. Verification / Alternative check: Example check: take x = 100 (even) and y = 1917 (odd). Then ( -1 )^100 = 1 and ( -1 )^1917 = -1, sum = 0. Any other even-odd pair gives the same result. Why Other Options Are Wrong: 2: would require both powers to be 1, meaning both exponents even, but then x + y would be even. -2: would require both powers to be -1, meaning both exponents odd, but then x + y would be even. 1 or -1: cannot happen because the two terms are always either (1,1), (-1,-1), or (1,-1). Common Pitfalls: Forgetting that an odd sum forces opposite parity, or assuming x and y can both be odd because 2017 is odd (actually odd + odd is even). Final Answer: 0

Question 10

If sqrt(a^2 + b^2 + a*b) + sqrt(a^2 + b^2 - a*b) = 1, what is the exact value of (1 - a^2) * (1 - b^2)?

Accepted Answer

Introduction / Context: This problem looks intimidating, but it becomes clean once we convert the given square-root expression into simpler helper variables. It tests algebraic manipulation, squaring safely, and using identities like (p + q)^2 and (p - q)^2 to extract u and v type quantities. Given Data / Assumptions: sqrt(a^2 + b^2 + a*b) + sqrt(a^2 + b^2 - a*b) = 1 Let u = a^2 + b^2 and v = a*b Concept / Approach: Introduce p and q as the two square-roots. Then rewrite u and v in terms of p and q. Finally compute (1 - a^2)(1 - b^2) = 1 - (a^2 + b^2) + a^2 b^2 = 1 - u + v^2. Step-by-Step Solution: Step 1: Let p = sqrt(u + v) and q = sqrt(u - v). Then p + q = 1. Step 2: Square p and q: p^2 = u + v and q^2 = u - v. Step 3: Add: p^2 + q^2 = 2u, so u = (p^2 + q^2)/2. Step 4: Subtract: p^2 - q^2 = 2v, so v = (p^2 - q^2)/2. Step 5: Use p + q = 1. Then u = ((p + q)^2 - 2pq)/2 = (1 - 2pq)/2 = 1/2 - pq. Step 6: Also v^2 = ((p^2 - q^2)/2)^2 = ((p - q)(p + q)/2)^2 = ((p - q)/2)^2. Step 7: Now compute E = 1 - u + v^2 = 1 - (1/2 - pq) + (p - q)^2/4. Step 8: Expand (p - q)^2 = (p + q)^2 - 4pq = 1 - 4pq. Step 9: So E = 1/2 + pq + (1 - 4pq)/4 = 1/2 + pq + 1/4 - pq = 3/4. Verification / Alternative check: Notice the pq terms cancel completely, meaning the value is constant (independent of specific a and b) as long as the given condition holds. Why Other Options Are Wrong: 1/4, 4/7, 5/4, 1: these do not match the invariant value obtained after proper transformation; the expression simplifies exactly to a constant 3/4. Common Pitfalls: Squaring without setting helper variables, mixing up u and v signs, or forgetting that (1 - a^2)(1 - b^2) = 1 - (a^2 + b^2) + a^2 b^2. Final Answer: 3/4

Question 11

If 3x + 4y - 11 = 18 and 8x - 6y + 12 = 6, what is the value of the expression 5x - 3y - 9?

Accepted Answer

Introduction / Context: This question tests solving simultaneous linear equations and then evaluating a new linear expression using the solved values. A clean approach is to first simplify both equations into standard form, solve for x and y, and substitute into 5x - 3y - 9. Given Data / Assumptions: 3x + 4y - 11 = 18 8x - 6y + 12 = 6 Find 5x - 3y - 9 Concept / Approach: Convert each equation into ax + by = c form. Then solve using substitution or elimination. Finally compute the required expression exactly. Step-by-Step Solution: Step 1: From 3x + 4y - 11 = 18, add 11 to both sides: 3x + 4y = 29. Step 2: From 8x - 6y + 12 = 6, subtract 12 from both sides: 8x - 6y = -6. Step 3: Divide Step 2 by 2 to simplify: 4x - 3y = -3. Step 4: Solve 4x - 3y = -3 for x: 4x = -3 + 3y, so x = (-3 + 3y)/4. Step 5: Substitute into 3x + 4y = 29: 3((-3 + 3y)/4) + 4y = 29. Step 6: Multiply by 4: 3(-3 + 3y) + 16y = 116. Step 7: Expand: -9 + 9y + 16y = 116 => -9 + 25y = 116 => 25y = 125 => y = 5. Step 8: Then x = (-3 + 15)/4 = 12/4 = 3. Step 9: Compute 5x - 3y - 9 = 5*3 - 3*5 - 9 = 15 - 15 - 9 = -9. Verification / Alternative check: Check in 3x + 4y = 29: 3*3 + 4*5 = 9 + 20 = 29 correct. Check 8x - 6y = -6: 24 - 30 = -6 correct. Why Other Options Are Wrong: -18, 18, -27, 9: these come from arithmetic slips (wrong substitution, sign error in moving constants, or wrong final evaluation). Common Pitfalls: Forgetting to move -11 and +12 correctly, or not simplifying 8x - 6y = -6 before substitution. Final Answer: -9

Question 12

If a + b + c = 7/12, 3a - 4b + 5c = 3/4, and 7a - 11b - 13c = -7/12, what is the value of a + c?

Accepted Answer

Introduction / Context: This question tests solving a system of three linear equations in three variables and then finding a required combination (a + c) without necessarily computing everything in the most complicated way. The key is to eliminate b using the first equation and reduce the system to two equations in a and c. Given Data / Assumptions: a + b + c = 7/12 3a - 4b + 5c = 3/4 7a - 11b - 13c = -7/12 Find a + c Concept / Approach: Express b from the first equation, substitute into the other two equations, solve the resulting two-equation system in a and c, then add a + c. Step-by-Step Solution: Step 1: From a + b + c = 7/12, write b = 7/12 - a - c. Step 2: Substitute into 3a - 4b + 5c = 3/4. Step 3: 3a - 4(7/12 - a - c) + 5c = 3/4. Step 4: 3a - 28/12 + 4a + 4c + 5c = 9/12 => 7a + 9c = 37/12. Step 5: Substitute b into 7a - 11b - 13c = -7/12. Step 6: 7a - 11(7/12 - a - c) - 13c = -7/12. Step 7: 7a - 77/12 + 11a + 11c - 13c = -7/12 => 18a - 2c = 35/6. Step 8: Divide by 2: 9a - c = 35/12 => c = 9a - 35/12. Step 9: Put c into 7a + 9c = 37/12: 7a + 9(9a - 35/12) = 37/12. Step 10: 7a + 81a - 315/12 = 37/12 => 88a = 352/12 => a = 1/3. Step 11: c = 9*(1/3) - 35/12 = 3 - 35/12 = 1/12. Step 12: a + c = 1/3 + 1/12 = 5/12. Verification / Alternative check: You can back-substitute to confirm all three equations are satisfied; the arithmetic matches exactly, so the derived a + c is reliable. Why Other Options Are Wrong: 1/2, 1/4, 3/4, 7/12: these typically occur from fraction handling mistakes (especially converting 3/4 to 9/12 and distributing negatives with -11b). Common Pitfalls: Sign errors while substituting b = 7/12 - a - c, or mixing denominators (12 and 6) without converting carefully. Final Answer: 5/12

Question 13

The ratio of curved surface area to total surface area of a right circular cylinder is 2:5. If the total surface area is 3080 cm^2, what is the volume of the cylinder (in cm^3)?

Accepted Answer

Introduction / Context: This question combines cylinder surface-area formulas with ratio reasoning. The ratio CSA:TSA = 2:5 immediately gives a relationship between height and radius, and then the given TSA lets us compute the exact radius. Finally, volume follows from V = pi*r^2*h. This is a classic multi-step mensuration problem. Given Data / Assumptions: CSA:TSA = 2:5 TSA = 3080 cm^2 CSA = 2*pi*r*h TSA = 2*pi*r*(h + r) Use pi = 22/7 (standard for such options) Concept / Approach: Use the ratio to connect CSA and TSA, then derive h in terms of r using TSA/CSA = (h + r)/h. Use CSA to find r, then compute volume. Step-by-Step Solution: Step 1: Since CSA:TSA = 2:5 and TSA = 3080, CSA = (2/5)*3080 = 1232. Step 2: TSA/CSA = 3080/1232 = 5/2. Step 3: TSA/CSA = (2*pi*r*(h + r)) / (2*pi*r*h) = (h + r)/h. Step 4: So (h + r)/h = 5/2 => 1 + (r/h) = 5/2 => r/h = 3/2 => h = (2/3)*r. Step 5: Use CSA = 2*pi*r*h = 2*pi*r*(2r/3) = (4/3)*pi*r^2 = 1232. Step 6: r^2 = 1232*3/(4*pi) = 924/pi. With pi = 22/7, r^2 = 924*7/22 = 294. Step 7: r = sqrt(294) = 7*sqrt(6). Then h = (2/3)*r = (14/3)*sqrt(6). Step 8: Volume V = pi*r^2*h = pi*294*(14/3)*sqrt(6) = pi*1372*sqrt(6). Step 9: With pi = 22/7, V = 1372*(22/7)*sqrt(6) = 4312*sqrt(6). Verification / Alternative check: Check TSA using r = 7*sqrt(6) and h = (2/3)r: TSA becomes 3080 cm^2 exactly (with pi = 22/7), confirming consistency. Why Other Options Are Wrong: Other coefficients (4522, 3822, 4642, 4112): come from wrong CSA value, wrong ratio inversion, or wrong pi usage. Common Pitfalls: Mixing CSA with TSA, forgetting the 2*pi*r factor cancels in the ratio, or using (2/5) incorrectly as (5/2). Final Answer: 4312*sqrt(6)

Question 14

In trigonometry, simplify and find the exact value of (cos 7A + cos 5A) / (sin 7A - sin 5A).

Accepted Answer

Introduction / Context: This question tests standard trigonometric sum-to-product identities. Expressions like cos C + cos D and sin C - sin D are designed so that the same common factor appears in numerator and denominator, allowing easy cancellation and producing a simple trig ratio (like tan or cot). Given Data / Assumptions: Expression: (cos 7A + cos 5A) / (sin 7A - sin 5A) Use identities: cos C + cos D = 2*cos((C + D)/2)*cos((C - D)/2) sin C - sin D = 2*cos((C + D)/2)*sin((C - D)/2) Concept / Approach: Convert both numerator and denominator into product form and cancel the common factor 2*cos((C + D)/2). Then simplify to a basic trig ratio. Step-by-Step Solution: Step 1: Numerator: cos 7A + cos 5A = 2*cos((7A + 5A)/2)*cos((7A - 5A)/2). Step 2: That becomes 2*cos(6A)*cos(A). Step 3: Denominator: sin 7A - sin 5A = 2*cos((7A + 5A)/2)*sin((7A - 5A)/2). Step 4: That becomes 2*cos(6A)*sin(A). Step 5: Divide: (2*cos(6A)*cos(A)) / (2*cos(6A)*sin(A)). Step 6: Cancel 2*cos(6A): result = cos(A)/sin(A) = cot A. Verification / Alternative check: Pick A = 10 degrees (any valid value): both sum-to-product forms remain valid and cancellation always leads to cot A, confirming the simplification is identity-based, not numeric coincidence. Why Other Options Are Wrong: tan A: inverse of the correct ratio. tan 4A or cot 4A: would appear if the half-sum or half-difference were 4A, but here they are 6A and A. sec A: not a quotient of sin and cos directly here. Common Pitfalls: Using the wrong identity for sin C - sin D (it uses cos of half-sum, not sin), or mixing up half-sum and half-difference. Final Answer: cot A

Question 15

Simplify the trigonometric expression: (1 - sin(90° - 2A)) / (1 + sin(90° + 2A)).

Accepted Answer

Introduction / Context: This question tests co-function identities involving 90 degrees and then the classic algebraic simplification of (1 - cos x) / (1 + cos x). After rewriting the sine terms, the expression collapses into a perfect square ratio of sine and cosine, giving tan^2 A. Given Data / Assumptions: Expression: (1 - sin(90° - 2A)) / (1 + sin(90° + 2A)) Use identities: sin(90° - x) = cos x sin(90° + x) = cos x 1 - cos 2A = 2*sin^2 A 1 + cos 2A = 2*cos^2 A Concept / Approach: Convert both sine terms into cos(2A). Then use double-angle identities to convert the numerator and denominator into squares of sin A and cos A and simplify the ratio. Step-by-Step Solution: Step 1: sin(90° - 2A) = cos(2A). Step 2: sin(90° + 2A) = cos(2A). Step 3: Expression becomes (1 - cos(2A)) / (1 + cos(2A)). Step 4: Use identities: 1 - cos(2A) = 2*sin^2 A and 1 + cos(2A) = 2*cos^2 A. Step 5: So the ratio is (2*sin^2 A) / (2*cos^2 A) = sin^2 A / cos^2 A. Step 6: sin^2 A / cos^2 A = (sin A / cos A)^2 = tan^2 A. Verification / Alternative check: Try A = 45 degrees: cos(90) = 0, so expression = (1 - 0)/(1 + 0) = 1, and tan^2 45 degrees = 1. Matches perfectly. Why Other Options Are Wrong: cot 2A or tan 2A: wrong angle and not squared. sin A * cos A: product form, not a ratio. cos^2 A: would require numerator to be cos^2 and denominator 1. Common Pitfalls: Incorrectly thinking sin(90° + x) becomes -cos x (it is +cos x), or forgetting the double-angle conversions. Final Answer: tan^2 A

Question 16

Compute the exact value of sin 75° + sin 15° using standard trigonometric identities.

Accepted Answer

Introduction / Context: This problem tests the sum-to-product identity for sine. Angles like 75 degrees and 15 degrees are chosen because their average is 45 degrees and their half-difference is 30 degrees, which have well-known exact sine and cosine values. The expression becomes a clean product. Given Data / Assumptions: Find sin 75° + sin 15° Use identity: sin C + sin D = 2*sin((C + D)/2)*cos((C - D)/2) Known values: sin 45° = 1/sqrt(2) cos 30° = sqrt(3)/2 Concept / Approach: Convert the sum of sines into a product, then substitute exact standard-angle values to get an exact simplified surd form. Step-by-Step Solution: Step 1: Apply identity with C = 75° and D = 15°. Step 2: sin 75° + sin 15° = 2*sin((75 + 15)/2)*cos((75 - 15)/2). Step 3: (75 + 15)/2 = 90/2 = 45° and (75 - 15)/2 = 60/2 = 30°. Step 4: So expression = 2*sin 45°*cos 30°. Step 5: Substitute values: = 2*(1/sqrt(2))*(sqrt(3)/2). Step 6: Cancel 2 and 2: = sqrt(3)/sqrt(2) = sqrt(3/2). Verification / Alternative check: Numerically, sin 75° is about 0.9659 and sin 15° is about 0.2588, sum about 1.2247. sqrt(3/2) is also about 1.2247, confirming the exact result. Why Other Options Are Wrong: sqrt(3), 2*sqrt(3): too large compared to the numeric sum. 3/sqrt(2): equals about 2.121, far too large. sqrt(2/3): about 0.816, too small. Common Pitfalls: Using the wrong identity (difference instead of sum) or halving angles incorrectly (mixing 60 and 30). Final Answer: sqrt(3/2)

Question 17

Simplify the expression: [(cos 3θ + 2cos 5θ + cos 7θ) / (cos θ + 2cos 3θ + cos 5θ)] + (sin 2θ * tan 3θ).

Accepted Answer

Introduction / Context: This is a structured trigonometric simplification problem. The numerator and denominator are symmetric in cosine terms (3θ, 5θ, 7θ and θ, 3θ, 5θ), suggesting sum-to-product grouping. After simplifying the fraction, the remaining sin 2θ * tan 3θ term neatly cancels a part of the cosine expansion, leaving a very simple final result. Given Data / Assumptions: Expression: [(cos 3θ + 2cos 5θ + cos 7θ)/(cos θ + 2cos 3θ + cos 5θ)] + sin 2θ * tan 3θ Use: cos X + cos Y = 2*cos((X + Y)/2)*cos((X - Y)/2) Use: tan 3θ = sin 3θ / cos 3θ Use: cos(3θ + 2θ) = cos 3θ*cos 2θ - sin 3θ*sin 2θ Concept / Approach: Group paired cosine terms (cos 3θ + cos 7θ) and (cos θ + cos 5θ) to factor out common multipliers. Reduce the fraction to cos 5θ / cos 3θ, then combine with sin 2θ*sin 3θ / cos 3θ and apply the cosine addition identity to collapse everything to cos 2θ. Step-by-Step Solution: Step 1: Group numerator: (cos 3θ + cos 7θ) + 2cos 5θ. Step 2: cos 3θ + cos 7θ = 2*cos 5θ*cos 2θ. Step 3: Numerator becomes 2*cos 5θ*cos 2θ + 2*cos 5θ = 2*cos 5θ*(cos 2θ + 1). Step 4: Similarly denominator: (cos θ + cos 5θ) + 2cos 3θ. Step 5: cos θ + cos 5θ = 2*cos 3θ*cos 2θ. Step 6: Denominator becomes 2*cos 3θ*cos 2θ + 2*cos 3θ = 2*cos 3θ*(cos 2θ + 1). Step 7: Fraction simplifies to (2*cos 5θ*(cos 2θ + 1)) / (2*cos 3θ*(cos 2θ + 1)) = cos 5θ / cos 3θ. Step 8: sin 2θ * tan 3θ = sin 2θ*(sin 3θ/cos 3θ) = (sin 2θ*sin 3θ)/cos 3θ. Step 9: Total = (cos 5θ + sin 2θ*sin 3θ) / cos 3θ. Step 10: Use cos(3θ + 2θ) = cos 3θ*cos 2θ - sin 3θ*sin 2θ, so cos 5θ = cos 3θ*cos 2θ - sin 3θ*sin 2θ. Step 11: Then cos 5θ + sin 2θ*sin 3θ = cos 3θ*cos 2θ. Step 12: Divide by cos 3θ: result = cos 2θ. Verification / Alternative check: Pick θ = 0: the expression becomes (1 + 2*1 + 1)/(1 + 2*1 + 1) + 0 = 1, and cos 0 = 1, so it matches. Why Other Options Are Wrong: sin 2θ, tan 2θ, sec 2θ, cot θ*sin 2θ: these do not arise after the clean cancellation with cos 3θ and the cos(3θ+2θ) identity. Common Pitfalls: Not grouping cosine pairs correctly, forgetting tan 3θ = sin 3θ / cos 3θ, or missing the cancellation of sin 2θ*sin 3θ terms. Final Answer: cos 2θ

Question 18

Simplify and find the exact value of [2 * sin(45° + θ) * sin(45° - θ)] / cos 2θ.

Accepted Answer

Introduction / Context: This problem tests product-to-sum identities and the ability to recognize a cancellation with cos 2θ. The angles (45° + θ) and (45° - θ) are conjugate pairs, so their product collapses to a simple cosine expression using sin u * sin v identity. The remaining division by cos 2θ completes the simplification. Given Data / Assumptions: Expression: [2*sin(45° + θ)*sin(45° - θ)]/cos 2θ Identity: sin u * sin v = (cos(u - v) - cos(u + v))/2 cos 90° = 0 Concept / Approach: Apply sin u * sin v identity, simplify u - v and u + v, then multiply by 2 and divide by cos 2θ. The structure is designed for exact cancellation. Step-by-Step Solution: Step 1: Let u = 45° + θ and v = 45° - θ. Step 2: sin u * sin v = (cos(u - v) - cos(u + v))/2. Step 3: u - v = (45° + θ) - (45° - θ) = 2θ. Step 4: u + v = (45° + θ) + (45° - θ) = 90°. Step 5: So sin u * sin v = (cos 2θ - cos 90°)/2 = (cos 2θ - 0)/2 = cos 2θ/2. Step 6: Multiply by 2: 2*sin u*sin v = 2*(cos 2θ/2) = cos 2θ. Step 7: Divide by cos 2θ: (cos 2θ)/(cos 2θ) = 1. Verification / Alternative check: Take θ = 0: numerator becomes 2*sin45*sin45 = 2*(1/sqrt(2))^2 = 1, and cos 0 = 1, ratio = 1. Matches the simplified result. Why Other Options Are Wrong: 0: would require the numerator to be 0 for all θ, which is not true. tan 2θ or cot 2θ: these would appear if the numerator became sin 2θ, but it becomes cos 2θ. sec 2θ: would be 1/cos 2θ, not 1. Common Pitfalls: Using the wrong product identity (confusing sin u sin v with sin u cos v), or miscomputing u + v as 2*45° + 2θ instead of 90°. Final Answer: 1

Question 19

Simplify and evaluate: sin(90° + 2A) * [4 - cos^2(90° - 2A)].

Accepted Answer

Introduction / Context: This question tests co-function identities, power manipulations, and recognizing higher-power patterns like cos^6 A - sin^6 A. The notation cos^2(90° - 2A) means the square of cos(90° - 2A). After converting the 90-degree shifts, the expression becomes a product in sin A and cos A that matches the factorization of cos^6 A - sin^6 A. Given Data / Assumptions: Expression: sin(90° + 2A) * [4 - cos^2(90° - 2A)] Identities: sin(90° + x) = cos x cos(90° - x) = sin x cos 2A = cos^2 A - sin^2 A sin 2A = 2*sin A*cos A Concept / Approach: Convert the shifted angles to basic sin and cos of 2A, rewrite everything in terms of sin A and cos A, and match it to 4*(cos^6 A - sin^6 A) using factorization: cos^6 - sin^6 = (cos^2 - sin^2)(cos^4 + cos^2 sin^2 + sin^4). Step-by-Step Solution: Step 1: sin(90° + 2A) = cos 2A. Step 2: cos(90° - 2A) = sin 2A, so cos^2(90° - 2A) = sin^2 2A. Step 3: Expression becomes cos 2A * [4 - sin^2 2A]. Step 4: Write sin^2 2A = (2*sin A*cos A)^2 = 4*sin^2 A*cos^2 A. Step 5: So [4 - sin^2 2A] = 4 - 4*sin^2 A*cos^2 A = 4*(1 - sin^2 A*cos^2 A). Step 6: Also cos 2A = cos^2 A - sin^2 A. Step 7: Expression = 4*(cos^2 A - sin^2 A)*(1 - sin^2 A*cos^2 A). Step 8: Note that (1 - sin^2 A*cos^2 A) = cos^4 A + cos^2 A*sin^2 A + sin^4 A (since (cos^2 + sin^2)^2 = 1 gives cos^4 + 2cos^2 sin^2 + sin^4 = 1, rearrange to get cos^4 + sin^4 = 1 - 2cos^2 sin^2, then add cos^2 sin^2 to get 1 - cos^2 sin^2). Step 9: Therefore expression = 4*(cos^2 A - sin^2 A)*(cos^4 A + cos^2 A*sin^2 A + sin^4 A) = 4*(cos^6 A - sin^6 A). Verification / Alternative check: Take A = 45°: cos^6 A - sin^6 A = 0, so RHS = 0. LHS: cos 90° = 0, so LHS = 0. Matches. Why Other Options Are Wrong: 4*(cos^6 A + sin^6 A): uses a plus, but the factor (cos^2 - sin^2) creates a difference. 2*(cos^3 A ± sin^3 A): wrong power structure. cos 2A: misses the large multiplier and squared term effect. Common Pitfalls: Misreading cos^2(90° - 2A) as cos(2*(90° - 2A)) instead of (cos(90° - 2A))^2, and failing to convert sin 2A correctly. Final Answer: 4*(cos^6 A - sin^6 A)

Question 20

Simplify the expression: [cos(90° + A) / sec(270° - A)] + [sin(270° + A) / cosec(630° - A)].

Accepted Answer

Introduction / Context: This question tests trigonometric angle-shift identities (90°, 270°, 630°) and reciprocal functions (sec, cosec). These angles are designed so that everything reduces to simple sin A and cos A terms. After simplification, the expression becomes sin^2 A + cos^2 A, which is always 1. Given Data / Assumptions: Expression: cos(90° + A)/sec(270° - A) + sin(270° + A)/cosec(630° - A) Identities: cos(90° + A) = -sin A sin(270° + A) = -cos A sec x = 1/cos x cosec x = 1/sin x cos(270° - A) = -sin A sin(630° - A) = sin(360° + 270° - A) = sin(270° - A) = -cos A Concept / Approach: Rewrite each trig function using standard shift identities, convert sec and cosec into cos and sin, then simplify each fraction separately and add. Step-by-Step Solution: Step 1: cos(90° + A) = -sin A. Step 2: cos(270° - A) = -sin A, so sec(270° - A) = 1/cos(270° - A) = 1/(-sin A) = -cosec A. Step 3: First term = (-sin A)/(-cosec A) = (-sin A)/(-1/sin A) = sin^2 A. Step 4: sin(270° + A) = -cos A. Step 5: sin(630° - A) = sin(270° - A) = -cos A, so cosec(630° - A) = 1/(-cos A) = -sec A. Step 6: Second term = (-cos A)/(-sec A) = (-cos A)/(-1/cos A) = cos^2 A. Step 7: Sum = sin^2 A + cos^2 A = 1. Verification / Alternative check: Take A = 0: first term becomes cos90/sec270 = 0/undefined? but through simplified identity form it becomes sin^2 0 = 0 and second term becomes cos^2 0 = 1, total 1 (valid where defined). The identity result remains 1 wherever the original expression is defined. Why Other Options Are Wrong: 0 or -1: would contradict the identity sin^2 A + cos^2 A = 1. tan A*sec A or 3*sec A: involve sec, but the sec factors cancel during simplification. Common Pitfalls: Mis-evaluating 270° shifts (sign errors), and forgetting that sec and cosec are reciprocals, not independent functions. Final Answer: 1

Simplification Questions