Express tan(A/2) in terms of cosec A and cot A only. Which of the following expressions is equal to tan(A/2)?

Introduction / Context:
This question checks your understanding of half-angle identities and your ability to rewrite trigonometric expressions in alternative forms. You must express tan(A/2) using only cosec A and cot A. Instead of directly memorising this specific form, you can derive it from the standard half-angle formulas for tangent and then rewrite the result using the definitions of cosec and cot in terms of sine and cosine.

Given Data / Assumptions:

• We need an expression for tan(A/2) in terms of cosec A and cot A.
• Angle A is in a domain where all functions exist and are finite.
• Standard half-angle identities for tangent hold: tan(A/2) = sin A / (1 + cos A) and tan(A/2) = (1 − cos A) / sin A.
• cosec A = 1 / sin A and cot A = cos A / sin A.

Concept / Approach:
Begin with the known identity tan(A/2) = (1 − cos A) / sin A. Because we want the answer in terms of cosec A and cot A, we rewrite 1/sin A as cosec A and cos A/sin A as cot A. This transformation reveals tan(A/2) as a simple difference involving cosec A and cot A. This avoids any guesswork and demonstrates how standard identities can be recombined in useful ways.

Step-by-Step Solution:
Use the half-angle identity: tan(A/2) = (1 − cos A) / sin A.Rewrite the denominator: dividing numerator and denominator by sin A is not necessary here; instead, express each term using cosec A and cot A.Observe that 1/sin A = cosec A and cos A/sin A = cot A.Now write (1 − cos A)/sin A as 1/sin A − cos A/sin A.This becomes cosec A − cot A.Therefore tan(A/2) = cosec A − cot A.

Verification / Alternative check:
We can check numerically for a convenient angle, such as A = 60°. Then A/2 = 30°. We know tan 30° = 1/√3. Compute the right-hand side: cosec 60° = 1/sin 60° = 2/√3, and cot 60° = cos 60°/sin 60° = (1/2)/(√3/2) = 1/√3. Then cosec A − cot A = 2/√3 − 1/√3 = 1/√3, which matches tan 30°. This confirms the identity tan(A/2) = cosec A − cot A.

Why Other Options Are Wrong:

• cosecA + cotA is the reciprocal of (cosec A − cot A) when multiplied appropriately, but it does not equal tan(A/2).
• secA − cotA and secA + cotA mix secant with cotangent and are not standard half-angle forms.
• tanA / 2 is simply half of tan A, which is not equal to tan(A/2) in general.

Common Pitfalls:

• Confusing tan(A/2) with tan A / 2 and choosing option e by mistake.
• Using the wrong half-angle identity or forgetting that there are multiple equivalent forms of tan(A/2).
• Failing to correctly substitute 1/sin A and cos A/sin A with cosec A and cot A.

Final Answer:
cosecA - cotA