If x is defined by x = 1 / (cosec A − cot A), then simplify x to an equivalent standard trigonometric expression (assuming all terms are defined).

Introduction / Context: This question tests algebraic simplification with trigonometric identities, especially the use of conjugates. Expressions like 1/(cosec A - cot A) are designed to be simplified by multiplying numerator and denominator by the conjugate (cosec A + cot A). A key identity makes the denominator collapse neatly: cosec^2 A - cot^2 A = 1. Recognizing and applying this identity quickly is a common aptitude technique in trigonometry.

Given Data / Assumptions:

• x = 1 / (cosec A - cot A) • Assume cosec A and cot A are defined and cosec A - cot A ≠ 0 • Required: simplify x into a standard expression

Concept / Approach: Multiply by the conjugate: 1/(cosec A - cot A) * (cosec A + cot A)/(cosec A + cot A). The denominator becomes: (cosec A - cot A)(cosec A + cot A) = cosec^2 A - cot^2 A. Use the fundamental identity: cosec^2 A = 1 + cot^2 A, which rearranges to: cosec^2 A - cot^2 A = 1. So the entire denominator becomes 1, leaving only the numerator cosec A + cot A.

Step-by-Step Solution: 1) Start: x = 1 / (cosec A - cot A) 2) Multiply numerator and denominator by the conjugate (cosec A + cot A): x = (cosec A + cot A) / [(cosec A - cot A)(cosec A + cot A)] 3) Use difference of squares: (cosec A - cot A)(cosec A + cot A) = cosec^2 A - cot^2 A 4) Use identity cosec^2 A = 1 + cot^2 A: cosec^2 A - cot^2 A = (1 + cot^2 A) - cot^2 A = 1 5) Therefore: x = (cosec A + cot A) / 1 = cosec A + cot A

Verification / Alternative check: Pick A = 45°: cosec 45° = √2 and cot 45° = 1. LHS: 1/(√2 - 1). Rationalizing gives (√2 + 1). RHS: cosec A + cot A = √2 + 1. Both match, confirming the simplification.

Why Other Options Are Wrong: • cosec A - cot A is the original denominator, not the simplified reciprocal form. • cosec^2 A + cot^2 A does not appear after conjugate cancellation and is not equal to the expression. • sec A + tan A is a different identity family. • 1 would only happen if cosec A + cot A = 1, not generally true.

Common Pitfalls: • Forgetting to multiply by the conjugate and trying to “split” the denominator incorrectly. • Using cosec^2 A - cot^2 A = 0 (wrong); it equals 1.

Final Answer: cosec A + cot A