If x = cot A / √(1 + cot^2 A), then x is equal to which basic trigonometric function of A?

Introduction / Context:
This problem tests your understanding of Pythagorean identities involving cot A and cosec A, and the relationship between different trigonometric functions. The expression cot A / √(1 + cot^2 A) can be simplified using the identity for cosec^2 A in terms of cot^2 A. Once simplified, it becomes equal to a standard trigonometric function, which you must identify from the options.

Given Data / Assumptions:

• x = cot A / √(1 + cot^2 A).
• Identity: cosec^2 A = 1 + cot^2 A.
• cot A = cos A / sin A.
• cosec A = 1 / sin A.

Concept / Approach:
Use the identity 1 + cot^2 A = cosec^2 A to simplify the denominator. This replaces the square root with cosec A. Then express cot A and cosec A in terms of sine and cosine, and simplify the ratio. The final expression reduces to cos A, which is one of the basic trigonometric functions of A.

Step-by-Step Solution:

Verification / Alternative check:
Choose a specific angle such as A = 30 degrees. Then cot 30° = √3 and cosec 30° = 2. Compute x from the original formula: x = √3 / √(1 + 3) = √3 / 2. Now check cos 30° = √3 / 2. The values match, confirming that x = cos A is correct.

Why Other Options Are Wrong:

Common Pitfalls:
Common errors include misusing the identity and writing 1 + cot^2 A = sec^2 A, which is incorrect. Another pitfall is not handling the square root correctly and forgetting that √(cosec^2 A) equals cosec A in the typical acute angle setting. Finally, errors may occur when manipulating fractions of trigonometric functions, so carefully writing every step helps avoid mistakes.

Final Answer:
cos A