In trigonometry, if cot A is given by the identity cot A = sin B / (1 - cos B), and both angles are such that all functions are defined, then what is the value of cot 2A expressed purely in terms of angle B?

Introduction / Context:
This trigonometry question checks whether you can recognise standard half angle identities and then use them to relate angles A and B. The goal is not to evaluate any numeric angle, but to simplify cot 2A in terms of B by using the given relationship between cot A and an expression containing sin B and cos B. Such questions are common in competitive exams because they test formula knowledge and symbolic manipulation together.

Given Data / Assumptions:

• cot A = sin B / (1 - cos B).
• We assume angles A and B lie in ranges where all trigonometric functions involved are defined.
• We need to find cot 2A expressed only in terms of B.

Concept / Approach:
The key observation is that sin B / (1 - cos B) is a standard half angle form. Using the identity for tan(B / 2): tan(B / 2) = (1 - cos B) / sin B, we can see that its reciprocal is sin B / (1 - cos B) = cot(B / 2). This means cot A equals cot(B / 2), so A and B / 2 are cotangent equal angles, differing only by multiples of 180 degrees. Under the usual principal acute angle setting, we can take A = B / 2. Then 2A equals B, and cot 2A becomes cot B directly.

Step-by-Step Solution:
Recall the half angle identity: tan(B / 2) = (1 - cos B) / sin B. Take the reciprocal to obtain: cot(B / 2) = sin B / (1 - cos B). Given cot A = sin B / (1 - cos B), we have cot A = cot(B / 2). Therefore, A and B / 2 differ by multiples of 180 degrees. In a typical aptitude context with principal values, take A = B / 2. Now compute 2A: 2A = 2 * (B / 2) = B. Hence, cot 2A = cot B.

Verification / Alternative check:
We can check by substituting a specific angle B where the expression is defined. For example, let B be an acute angle such that the expression is valid. Then A = B / 2 gives cot A exactly equal to sin B / (1 - cos B) by the half angle identity. Doubling A returns B, so cot 2A is simply cot B. Because the relationship is derived from identities which hold for all admissible B, the conclusion cot 2A = cot B is generally valid in this context.

Why Other Options Are Wrong:
cot(B / 2) would represent cot A, not cot 2A. cot 2B and tan B introduce the wrong multiple or reciprocal of the angle. tan(B / 2) is the reciprocal of cot(B / 2), not cot 2A. Only cot B correctly matches the simplified form of cot 2A under the given relationship.

Common Pitfalls:
One common mistake is to try to express everything in terms of sin A and cos A instead of noticing the direct half angle identity in B. Another is to incorrectly double B and conclude cot 2A = cot 2B, which ignores that A itself is B / 2. Always identify known identities first before performing algebraic manipulations.

Final Answer:
The value of cot 2A in terms of B is cot B.