If cosec A + cot A = x for an acute angle A, then x can be expressed in which of the following equivalent trigonometric forms?

Introduction / Context:
This trigonometry question examines your understanding of identities involving sums and differences of cosecant, cotangent, secant, and tangent. Given that x = cosec A + cot A, you must find an equivalent expression for x in reciprocal form. Recognising the identity involving the product (cosec A + cot A)(cosec A − cot A) is the key to solving this problem quickly and accurately.

Given Data / Assumptions:

• cosec A + cot A = x.
• A is such that all involved trigonometric functions are defined.
• We are to express x using a reciprocal expression from the options.
• Standard Pythagorean identities for trigonometric functions hold.

Concept / Approach:
Recall the identity cosec^2 A − cot^2 A = 1, which is analogous to sec^2 A − tan^2 A = 1. Notice that (cosec A + cot A)(cosec A − cot A) expands to cosec^2 A − cot^2 A. If we use the identity cosec^2 A − cot^2 A = 1, then this product equals 1. Therefore, cosec A + cot A is the reciprocal of cosec A − cot A. This is the central idea that leads to the correct option.

Step-by-Step Solution:
Start from the product: (cosec A + cot A)(cosec A − cot A).Expand using the difference of squares: cosec^2 A − cot^2 A.Use the Pythagorean identity: cosec^2 A − cot^2 A = 1.Therefore (cosec A + cot A)(cosec A − cot A) = 1.Rearrange to express cosec A + cot A: cosec A + cot A = 1 / (cosec A − cot A).Since x = cosec A + cot A, we get x = 1 / (cosec A − cot A).

Verification / Alternative check:
As a quick numerical check, choose a simple angle such as A = 45°. Then sin 45° = cos 45° = √2/2, so cosec 45° = sec 45° = √2 and cot 45° = tan 45° = 1. Compute x = cosec A + cot A = √2 + 1. Now compute 1 / (cosec A − cot A) = 1 / (√2 − 1). Multiply numerator and denominator by (√2 + 1) to rationalise: 1 / (√2 − 1) = (√2 + 1) / ((√2 − 1)(√2 + 1)) = (√2 + 1) / (2 − 1) = √2 + 1. This matches x, confirming the identity.

Why Other Options Are Wrong:

• 1/(sec A − tan A) equals sec A + tan A, not cosec A + cot A, by a similar identity involving sec and tan.
• 1/(sec A − cos A) and 1/(sin A − cos A) do not simplify to x in any standard identity.
• 1/(sec A + tan A) is the reciprocal of sec A + tan A, not of cosec A + cot A.

Common Pitfalls:

• Confusing the identity cosec^2 A − cot^2 A = 1 with sec^2 A − tan^2 A = 1 and misapplying it.
• Assuming that 1/(sec A − tan A) might equal cosec A + cot A just by analogy without checking.
• Forgetting to expand the product carefully and verify that it reduces to 1.

Final Answer:
1/(cosecA - cotA)