In trigonometry, if cot(A/2) = x for an acute angle A, then which of the following expressions in terms of standard trigonometric functions of A is exactly equal to x?

Introduction / Context:
This problem tests your understanding of trigonometric identities, especially the half angle identities involving cot(A/2). Such questions are common in aptitude and competitive exams, where you must connect a given expression like cot(A/2) to different but equivalent forms written with sine, cosine, secant, and cosecant. The key idea is to recognise standard identities and then manipulate them algebraically to match one of the given options. This also checks your fluency with reciprocal and complementary relationships between the basic trigonometric functions.

Given Data / Assumptions:
- A is an acute angle, so all basic trigonometric functions of A are defined and have appropriate signs.
- cot(A/2) is denoted by x, and we want to express x in terms of functions of A only.
- We must select the expression that is exactly equal to cot(A/2).

Concept / Approach:
The standard half angle identity for cot(A/2) is:
cot(A/2) = (1 + cos A) / sin A.
Another very useful identity is the product identity:
(cosec A - cot A)(cosec A + cot A) = 1.
From this, we get cosec A + cot A = 1 / (cosec A - cot A). If we can relate cot(A/2) to cosec A + cot A, we can then connect it to 1 / (cosec A - cot A).

Step-by-Step Solution:
Step 1: Start from the half angle formula cot(A/2) = (1 + cos A) / sin A. Step 2: Rewrite the right hand side as a sum of two fractions: (1 + cos A) / sin A = 1/sin A + cos A/sin A. Step 3: Recognise that 1/sin A is cosec A and cos A/sin A is cot A. Step 4: Therefore cot(A/2) = cosec A + cot A. Step 5: Use the identity (cosec A - cot A)(cosec A + cot A) = 1, which rearranges to cosec A + cot A = 1 / (cosec A - cot A). Step 6: Substitute this result into the earlier expression to obtain cot(A/2) = 1 / (cosec A - cot A). Step 7: Compare this final expression with the given options and note that option d is an exact match.
Verification / Alternative check:
You can verify the identity numerically by taking an example, say A = 60 degrees. Compute cot(A/2) = cot(30 degrees) = √3. Then evaluate 1 / (cosec A - cot A) at A = 60 degrees and confirm that the numerical value also equals √3. This quick numeric check supports the algebraic derivation and reduces the risk of sign or reciprocal errors.

Why Other Options Are Wrong:
Option a, tan A / (1 + sec A), uses tangent and secant but does not simplify to the half angle cot(A/2).
Option b, 1 / (sec A + cot A), mixes secant and cotangent in a way that does not match the identities for cot(A/2).
Option c, tan A / (1 + cosec A), also does not connect correctly to the standard half angle formulas and gives a different value.
Option e, cosec A - cot A, is actually the reciprocal of the correct expression and therefore is not equal to x.

Common Pitfalls:
Many students confuse the formulas for tan(A/2) and cot(A/2) or mix up the sign in the identities involving cosec A and cot A. Another common mistake is to invert the wrong quantity or forget that (cosec A - cot A)(cosec A + cot A) = 1, which leads to the reciprocal relationship used here. Careless algebraic manipulation or skipping intermediate steps can also produce sign errors. Writing out each step clearly helps to avoid these traps.

Final Answer:
The correct expression for x is 1 / (cosec A - cot A).