If sin x = 4/5 for an acute angle x in a right triangle, find the value of cosec x + cot x.

Introduction / Context:
This trigonometry question tests the ability to move between the basic trigonometric ratios sin, cosec and cot using a right triangle. When one ratio is known, the others can be found by using Pythagoras theorem and the definitions of each function. The angle x is acute, so all primary trigonometric functions are positive, which simplifies the reasoning.

Given Data / Assumptions:

• sin x = 4/5 for an acute angle x.

• x lies in the first quadrant, so sin x, cos x, cosec x and cot x are all positive.

• We need to calculate cosec x + cot x in exact fractional form.

Concept / Approach:
For any right triangle with acute angle x, sin x = opposite / hypotenuse and cos x = adjacent / hypotenuse. If sin x is known, we can reconstruct the triangle sides in a convenient ratio and then compute cos x, cosec x and cot x. The reciprocal relation gives cosec x = 1 / sin x. The cotangent function is cot x = cos x / sin x. Substituting the specific values and simplifying yields the required sum.

Step-by-Step Solution:
Step 1: Interpret sin x = 4/5 as opposite side = 4k and hypotenuse = 5k for some positive k. Step 2: Use Pythagoras theorem to find the adjacent side: adjacent^2 = hypotenuse^2 − opposite^2 = (5k)^2 − (4k)^2 = 25k^2 − 16k^2 = 9k^2. Step 3: Therefore adjacent side = 3k. Step 4: Compute cosec x. Since cosec x = 1 / sin x, we get cosec x = 1 / (4/5) = 5/4. Step 5: Compute cot x. Since cot x = cos x / sin x and cos x = adjacent / hypotenuse = 3k / 5k = 3/5, we have cot x = (3/5) / (4/5) = 3/4. Step 6: Add the two values: cosec x + cot x = 5/4 + 3/4 = 8/4 = 2.

Verification / Alternative check:
We can verify numerically by assigning a specific triangle with sides 3, 4 and 5. Then sin x = 4/5 is consistent with the given condition. For this triangle, cosec x = 5/4 and cot x = 3/4, so their sum is 2. Any scaling of the triangle by factor k keeps the ratios unchanged, so the result is independent of k. This confirms that the exact answer 2 is correct and not an approximation.

Why Other Options Are Wrong:
Option 31/12 and 35/12 are larger than 2 and would arise from incorrect side lengths or algebraic errors when adding fractions. Option 1/2 is far too small and would require both cosec x and cot x to be less than 1, which cannot happen when sin x = 4/5. Option 9/5 equals 1.8 and could appear if a candidate misreads cot x as 1/tan x without computing properly. None of these match the correct, systematically derived value.

Common Pitfalls:
Some learners confuse cot x with tan x, leading to inverted ratios. Others forget to apply Pythagoras theorem correctly and may mistakenly treat the hypotenuse as the longest side without checking the arithmetic. Errors in fraction addition, such as adding numerators but keeping the same denominator, are also common. Drawing the triangle, labelling sides, and writing each trig ratio explicitly helps avoid these mistakes and reinforces conceptual understanding of right triangle trigonometry.

Final Answer:
The required value of cosec x + cot x is 2.