If cot(A/2) = x in trigonometry, then which of the following expressions is equal to x?

Introduction / Context:
This question focuses on half angle identities for the cotangent function. Relations between cot(A/2), cosec A, and cot A are important for simplifying trigonometric expressions and solving certain types of equations in trigonometry.

Given Data / Assumptions:

• cot(A/2) = x
• A is an angle for which the functions are defined.
• We must express x in terms of cosec A and cot A.

Concept / Approach:
One standard half angle identity for cotangent is: cot(A/2) = cosec A + cot A This follows from expressing sin A and cos A in terms of half angles and then simplifying. The identity provides a direct link between cot(A/2) and functions of A.

Step-by-Step Solution:
We know tan(A/2) = sin A / (1 + cos A) or (1 - cos A) / sin A The reciprocal relation is cot(A/2) = (1 + cos A) / sin A Now write (1 + cos A) / sin A as 1/sin A + cos A/sin A 1/sin A equals cosec A cos A/sin A equals cot A Therefore, cot(A/2) = cosec A + cot A So x = cosec A + cot A

Verification / Alternative check:
Take a specific angle, for example A = 60 degrees. Then A/2 = 30 degrees. Left side: cot 30° = √3. Right side: cosec 60° + cot 60°. We know cosec 60° = 1/(√3/2) = 2/√3 and cot 60° = 1/√3. Sum is 2/√3 + 1/√3 = 3/√3 = √3. Both sides match, supporting the identity.

Why Other Options Are Wrong:
Option a, cosec A - cot A, is the conjugate form and is sometimes used in manipulations but is not equal to cot(A/2). Options c and d involve sec A, which is unrelated to the standard half angle identity for cotangent. Option e, sec A + tan A, is linked to other identities but not to cot(A/2) in this direct way.

Common Pitfalls:
Students may confuse formulas for tan(A/2) with those for cot(A/2), or forget the reciprocal relationship. Another common error is incorrect conversion between sin and cosec or between cos and sec. Always derive cot(A/2) carefully from the known tan(A/2) identities to avoid memorization mistakes.

Final Answer:
The correct expression equal to x is cosec A + cot A.