Which of the following expressions is identically equal to 1 / (tan A + cot A) for an angle A where all trigonometric functions are defined?

Introduction / Context:
This question tests algebraic manipulation of trigonometric expressions and the ability to recognise useful identities. We are asked to find an expression that equals 1 / (tan A + cot A). The options involve combinations of cosec, sin, sec, and cos. The key is to rewrite everything in terms of sin A and cos A and simplify step by step.

Given Data / Assumptions:

• We must find an expression equal to 1 / (tan A + cot A).
• Angle A is such that all functions tan A, cot A, sin A, cos A, sec A, and cosec A are defined and non zero where used.
• Options involve products or square roots of expressions built from cosecA, sinA, secA, and cosA.
• We look for an identity that holds for all such angles A.

Concept / Approach:
First simplify tan A + cot A in terms of sine and cosine. We have tan A = sin A / cos A and cot A = cos A / sin A. Thus tan A + cot A = (sin^2 A + cos^2 A) / (sin A cos A) = 1 / (sin A cos A). Therefore 1 / (tan A + cot A) = sin A cos A. Next, we simplify each option in terms of sin A and cos A and see which one is equal to sin A cos A. We expect the correct option to give exactly sin A cos A after simplification.

Step-by-Step Solution:
Simplify the target: tan A + cot A = sin A / cos A + cos A / sin A = (sin^2 A + cos^2 A) / (sin A cos A) = 1 / (sin A cos A).Thus 1 / (tan A + cot A) = sin A cos A.Now examine option d: (cosecA - sinA)(secA - cosA).Rewrite in sin and cos: cosecA = 1 / sin A and secA = 1 / cos A.So (cosecA - sinA) = (1 / sin A - sin A) = (1 - sin^2 A) / sin A = cos^2 A / sin A.Similarly (secA - cosA) = (1 / cos A - cos A) = (1 - cos^2 A) / cos A = sin^2 A / cos A.Multiply them: (cos^2 A / sin A) * (sin^2 A / cos A) = (cos^2 A sin^2 A) / (sin A cos A) = sin A cos A.Therefore option d equals sin A cos A, which matches 1 / (tan A + cot A).

Verification / Alternative check:
We can test using a specific angle, for example A = 45 degrees. Then tan 45° = 1 and cot 45° = 1, so tan A + cot A = 2, and 1 / (tan A + cot A) = 1/2. Compute option d at A = 45 degrees. Sin 45° = cos 45° = sqrt(2)/2, cosec 45° = sec 45° = sqrt(2). Then cosecA - sinA = sqrt(2) - sqrt(2)/2 = sqrt(2)/2, and secA - cosA is the same, sqrt(2)/2. Their product is (sqrt(2)/2)^2 = 1/2, matching 1 / (tan 45° + cot 45°). This confirms the identity works at least for one test angle and together with algebraic simplification shows it is generally valid.

Why Other Options Are Wrong:

• Option a gives a different product; it simplifies to 1 / (sin A cos A), not sin A cos A.
• Options b and c involve square roots and thus represent the square root of some product, not the product itself.
• Option e changes the sign before cosA and does not simplify to sin A cos A.
• Only option d simplifies exactly to sin A cos A, which is 1 / (tan A + cot A).

Common Pitfalls:

• Not rewriting cosec and sec in terms of sin and cos and instead trying to manipulate them directly.
• Making algebraic mistakes when simplifying the products or forgetting to use sin^2 A + cos^2 A = 1.
• Confusing 1 / (sin A cos A) with sin A cos A, which are reciprocals.

Final Answer:
(cosecA - sinA)(secA - cosA)