What is the simplified value of the trigonometric expression cosec 2A + cot 2A in terms of the angle A?

Introduction / Context:
This question tests knowledge of a standard trigonometric identity that connects the sum cosec x + cot x with half angles. Recognizing this identity allows us to simplify expressions involving double angles quickly. Such identities are extremely useful in solving trigonometric equations and proving other results.

Given Data / Assumptions:

Expression: cosec 2A + cot 2A.
A is an angle for which the required trigonometric functions are defined and finite.
We must express the result in terms of A using basic identities.

Concept / Approach:
There is a well known identity: cosec x + cot x = cot(x / 2). This can be derived from half angle formulas or from geometric considerations in a right triangle. If we set x = 2A, then cosec 2A + cot 2A becomes cot A directly. Using this identity is the fastest route, but we can also verify it using sine and cosine definitions of the functions.

Step-by-Step Solution:
Recall that cosec x = 1 / sin x and cot x = cos x / sin x.Therefore cosec x + cot x = 1 / sin x + cos x / sin x = (1 + cos x) / sin x.Now use the half angle formulas: 1 + cos x = 2 cos^2(x / 2) and sin x = 2 sin(x / 2) cos(x / 2).Substitute into (1 + cos x) / sin x to get [2 cos^2(x / 2)] / [2 sin(x / 2) cos(x / 2)].Simplify by cancelling 2 and cos(x / 2): we obtain cos(x / 2) / sin(x / 2).But cos(x / 2) / sin(x / 2) is exactly cot(x / 2).Hence cosec x + cot x = cot(x / 2).Set x = 2A. Then cosec 2A + cot 2A = cot(2A / 2) = cot A.

Verification / Alternative check:
Take a specific angle, say A = 30°. Then 2A = 60°. Compute the left side: cosec 60° = 1 / (√3/2) = 2/√3, cot 60° = 1/√3, so their sum is 2/√3 + 1/√3 = 3/√3 = √3. On the right side, cot A = cot 30° = √3. Both sides match, giving confidence in the identity and the result.

Why Other Options Are Wrong:
The expressions sec A, sec(A/2), cot^2 A, and cosec A do not follow from the identity for cosec x + cot x. In particular, none of them can be simplified to (1 + cos 2A) / sin 2A, whereas cot A can. Therefore only cot A is consistent with the algebraic derivation.

Common Pitfalls:
Students sometimes confuse the identity cosec x + cot x = cot(x / 2) with similar looking identities for tan or sec, or they incorrectly try to express everything in terms of tan A without using half angles. Remembering and practicing standard half angle identities helps avoid such confusion.

Final Answer:
The simplified value of cosec 2A + cot 2A is cot A.