Simplify the trigonometric expression (secA - tanA)/(cosecA + cotA). The result is equal to which of the following?

Introduction / Context: This is a trigonometric simplification question that involves secant, tangent, cosecant and cotangent. Such expressions often appear in aptitude and board examination questions to test a deep understanding of reciprocal and quotient identities, as well as the ability to perform algebraic manipulations with trigonometric functions.

Given Data / Assumptions:

• The given expression is (secA - tanA)/(cosecA + cotA).
• We must find an equivalent expression from the options.
• A is an acute angle such that all trigonometric ratios involved are defined.

Concept / Approach: We use definitions of secA, tanA, cosecA and cotA in terms of sine and cosine. A helpful identity is that (secA - tanA)(secA + tanA) = 1 and similarly (cosecA - cotA)(cosecA + cotA) = 1. From these, we can solve for secA - tanA and cosecA + cotA in terms of their counterparts and then simplify the ratio. This approach reveals a neat reciprocal style identity connecting the numerator and the denominator.

Step-by-Step Solution: Step 1: Recall that secA = 1/cosA, tanA = sinA/cosA, cosecA = 1/sinA and cotA = cosA/sinA. Step 2: Use the identity (secA - tanA)(secA + tanA) = sec^2A - tan^2A. Step 3: Using sec^2A - tan^2A = 1, we get (secA - tanA)(secA + tanA) = 1. Step 4: Therefore secA - tanA = 1 / (secA + tanA). Step 5: Similarly, (cosecA - cotA)(cosecA + cotA) = cosec^2A - cot^2A = 1, so cosecA + cotA = 1 / (cosecA - cotA). Step 6: Substitute these results into the expression (secA - tanA)/(cosecA + cotA). Step 7: The expression becomes [1/(secA + tanA)] divided by [1/(cosecA - cotA)]. Step 8: Dividing by a fraction is the same as multiplying by its reciprocal, so the result is (1/(secA + tanA)) * (cosecA - cotA) = (cosecA - cotA)/(secA + tanA).

Verification / Alternative check: Take a specific angle such as A = 45 degrees. Compute sec45 degrees = √2, tan45 degrees = 1, cosec45 degrees = √2 and cot45 degrees = 1. The original expression becomes (√2 - 1)/(√2 + 1). The candidate expression (cosecA - cotA)/(secA + tanA) becomes (√2 - 1)/(√2 + 1), exactly the same. The other alternatives do not produce this value when evaluated numerically.

Why Other Options Are Wrong:

• Option A: (secA + tanA)/(cosecA + cotA) is essentially the reciprocal of the correct form and does not match the derived identity.
• Option B: (secA - tanA)/(cosecA - cotA) mixes both minus signs and does not make use of the reciprocal identity correctly.
• Option D: (secA + cosA)/(cosecA - sinA) imports sine and cosine terms that are unrelated to the neat reciprocal structure of the original expression.
• Option E: (secA - cosA)/(cosecA + sinA) similarly introduces unrelated combinations and fails numerical checks.

Common Pitfalls: A common error is to attempt direct substitution into sine and cosine without noticing the useful pattern (secA - tanA)(secA + tanA) = 1. This can lead to very messy algebra. Another mistake is to cancel terms incorrectly or to forget that division by a fraction requires multiplication by its reciprocal. Recognising and using the identity that the product of conjugate pairs equals 1 greatly simplifies the work.

Final Answer: Therefore, the simplified expression is (cosecA - cotA)/(secA + tanA).