If (1 - cosA)/(1 + cosA) = x, which equivalent expression in terms of cotA and cosecA correctly represents x?

Introduction / Context:
This question tests knowledge of trigonometric identities that connect cosine with cotangent and cosecant. Expressing (1 - cosA)/(1 + cosA) in different trigonometric forms is a standard skill in simplification problems that appear frequently in aptitude and competitive examinations.

Given Data / Assumptions:

• (1 - cosA)/(1 + cosA) = x
• A is an acute angle, so all trigonometric ratios are well defined
• We must represent x using cotA and cosecA

Concept / Approach:
The key idea is to express everything in terms of sine and cosine, then substitute definitions of cotA and cosecA. We know that cosecA = 1/sinA and cotA = cosA/sinA. There is also a known identity: (1 - cosA)/(1 + cosA) = (cosecA - cotA)^2. This identity can be derived systematically by writing cosecA and cotA in terms of sine and cosine and simplifying the square.

Step-by-Step Solution:
Step 1: Start with the expression (cosecA - cotA)^2. Step 2: Substitute cosecA = 1/sinA and cotA = cosA/sinA. Step 3: Then cosecA - cotA = (1/sinA) - (cosA/sinA) = (1 - cosA)/sinA. Step 4: Square this result: (cosecA - cotA)^2 = [(1 - cosA)/sinA]^2 = (1 - cosA)^2 / sin^2A. Step 5: Replace sin^2A with 1 - cos^2A, giving (1 - cosA)^2 / (1 - cos^2A). Step 6: Factor the denominator: 1 - cos^2A = (1 - cosA)(1 + cosA). Step 7: Cancel one factor of (1 - cosA) from numerator and denominator to obtain (1 - cosA)/(1 + cosA). Step 8: Therefore (cosecA - cotA)^2 = (1 - cosA)/(1 + cosA) = x.

Verification / Alternative check:
Take A = 60 degrees as a check. Then cos60 degrees = 1/2. Compute the left side: (1 - 1/2)/(1 + 1/2) = (1/2)/(3/2) = 1/3. Now compute (cosec60 degrees - cot60 degrees)^2. Here sin60 degrees = √3/2 so cosec60 degrees = 2/√3, and cot60 degrees = 1/√3. Their difference is 1/√3 and squaring gives 1/3, which matches the earlier result.

Why Other Options Are Wrong:

• Option A: (cotA + cosecA)^2 produces (1 + cosA)/(1 - cosA), the reciprocal ratio.
• Option C: cotA - cosecA is missing the square, so its value is different.
• Option D: cotA + cosecA has the wrong sign and no square.
• Option E: (1 + sinA)/(1 - sinA) corresponds to a different identity involving secA and tanA.

Common Pitfalls:
Learners often confuse (1 - cosA)/(1 + cosA) with its reciprocal (1 + cosA)/(1 - cosA). Another common error is to forget to square the binomial expression when converting it into forms involving cosecA and cotA. Careful algebra and deliberate factorisation avoid these mistakes.

Final Answer:
So the correct expression for x is (cotA - cosecA)^2.