If 1 / √(cosec^2 A − 1) = x for an acute angle A in trigonometry, then x is equal to which standard trigonometric ratio of A?

Introduction / Context:
This problem checks your understanding of basic trigonometric identities, especially the relation between cosec A and cot A. Questions of this type often appear in aptitude and competitive exams where you must quickly convert an expression involving one trigonometric function into another simpler function of the same angle.

Given Data / Assumptions:

• 1 / √(cosec^2 A − 1) = x.
• A is an acute angle, so 0° < A < 90°.
• We are asked to express x in terms of a basic trigonometric ratio of A.

Concept / Approach:
Key trigonometric identities used:

• cosec^2 A − 1 = cot^2 A.
• For an acute angle, square roots of squared trigonometric ratios are positive: √(cot^2 A) = cot A.
• If x = 1 / cot A, then x = tan A, because tan A is the reciprocal of cot A.

Step-by-Step Solution:
Start with the given expression: x = 1 / √(cosec^2 A − 1). Use the identity cosec^2 A − 1 = cot^2 A. Substitute this into the square root: √(cosec^2 A − 1) = √(cot^2 A). For an acute angle, √(cot^2 A) = cot A (positive value). Therefore x = 1 / cot A. Since tan A is the reciprocal of cot A, we have x = tan A.

Verification / Alternative check:
Take a simple acute angle, for example A = 30°. Then cosec 30° = 2, so cosec^2 30° = 4. Now cosec^2 30° − 1 = 3. Hence x = 1 / √3. But tan 30° is also 1 / √3, confirming that x = tan A is correct for this sample value. Because trigonometric identities are general, this holds for all acute angles.

Why Other Options Are Wrong:
Option b: cot A is the denominator inside the reciprocal and therefore cannot equal x.
Option c: sin A is unrelated to the identity cosec^2 A − 1 = cot^2 A in this context.
Option d: cos A is also not directly obtained from this identity.
Option e: sec A is the reciprocal of cos A, not of cot A, so it does not match the simplified value.

Common Pitfalls:
Students sometimes confuse the identity cosec^2 A − 1 = cot^2 A with sec^2 A − 1 = tan^2 A. Another common error is ignoring the fact that A is acute, which ensures that the square root remains positive. Correctly recognising and applying the identity is crucial to arrive at the right trigonometric ratio.

Final Answer:
The value of x is tan A.