If 1 / (tan A + cot A) = x, express x in terms of the trigonometric functions of angle A.

Introduction / Context:
This question checks your understanding of how to manipulate expressions involving tan A and cot A and convert them into equivalent forms involving sine and cosine. Such simplifications are common in trigonometry based aptitude and competitive exams.

Given Data / Assumptions:

- Expression: 1 / (tan A + cot A)
- This value is equal to x
- A is an angle where all involved functions are defined
- We want to write x using basic trigonometric functions of A

Concept / Approach:
The idea is to express tan A and cot A in terms of sine and cosine. We use tan A = sin A / cos A and cot A = cos A / sin A. After substituting, we combine the terms in the denominator, simplify the resulting fraction, and observe that the final expression reduces to a simple product of sine and cosine.

Step-by-Step Solution:
Step 1: Start from x = 1 / (tan A + cot A).Step 2: Replace tan A with sin A / cos A and cot A with cos A / sin A. The denominator becomes sin A / cos A + cos A / sin A.Step 3: Combine the two fractions in the denominator over a common denominator sin A cos A: (sin^2 A + cos^2 A) / (sin A cos A).Step 4: Use the fundamental identity sin^2 A + cos^2 A = 1. The denominator simplifies to 1 / (sin A cos A).Step 5: Thus x = 1 divided by [1 / (sin A cos A)] = sin A cos A.

Verification / Alternative check:
Pick an easy angle, for example A = 45 degrees. Then tan 45° = 1 and cot 45° = 1. So 1 / (tan 45° + cot 45°) = 1 / (1 + 1) = 1/2. Now compute sin A cos A at 45 degrees: sin 45° = √2 / 2 and cos 45° = √2 / 2, so sin A cos A = (√2 / 2) * (√2 / 2) = 2 / 4 = 1/2. Both match, confirming that x = sin A cos A.

Why Other Options Are Wrong:
- sin^2 A cos^2 A is the square of sin A cos A, not equal to it.
- sec A cosec A is 1 / (sin A cos A), which is the reciprocal of sin A cos A, not equal to x.
- sec^2 A cosec^2 A is even further from the simplified form and grows much faster in value.
- tan A + cot A is the original denominator, not its reciprocal.

Common Pitfalls:
A common error is to add tan A and cot A directly without using a common denominator or to misapply the identity sin^2 A + cos^2 A = 1. Another mistake is forgetting that we are taking the reciprocal of the entire sum, not the reciprocal of each term separately. Always convert to sine and cosine, combine carefully, and then simplify.

Final Answer:
The value of x is sin A cos A.