In trigonometric simplification, if X = sec^2(A) + cosec^2(A), express X in terms of tan(A) and cot(A) and select the correct equivalent expression for X (for angles where all terms are defined).

Introduction / Context:
This question tests your understanding of basic trigonometric identities and how to rewrite expressions involving secant and cosecant in terms of tangent and cotangent. Rewriting expressions in different trigonometric forms is very useful for simplification, solving equations, and proving identities in both school-level mathematics and aptitude exams.

Given Data / Assumptions:

• X is defined as X = sec^2(A) + cosec^2(A).
• A is an angle for which all the trigonometric functions sec(A), cosec(A), tan(A), and cot(A) are defined.
• We must choose the expression written in terms of tan(A) and cot(A) that is exactly equal to X.

Concept / Approach:
We use the fundamental identities that relate secant and cosecant to tangent and cotangent. Specifically, sec^2(A) = 1 + tan^2(A) and cosec^2(A) = 1 + cot^2(A). By substituting these into the definition of X, we obtain X in a new form purely in terms of tan(A) and cot(A). Then we compare the resulting expression to the given options and pick the matching one. This approach relies only on standard identities and straightforward algebraic simplification.

Step-by-Step Solution:
1) Start with the definition X = sec^2(A) + cosec^2(A). 2) Use the identity sec^2(A) = 1 + tan^2(A). 3) Use the identity cosec^2(A) = 1 + cot^2(A). 4) Substitute into X: X = [1 + tan^2(A)] + [1 + cot^2(A)]. 5) Combine like terms: X = 1 + tan^2(A) + 1 + cot^2(A) = 2 + tan^2(A) + cot^2(A). 6) Therefore the required expression for X in terms of tan(A) and cot(A) is X = 2 + tan^2(A) + cot^2(A).

Verification / Alternative check:
You can verify this identity numerically by choosing a specific angle A for which all functions are defined, for example A = 45 degrees. Then tan(45°) = 1 and cot(45°) = 1, so the right side becomes 2 + 1^2 + 1^2 = 4. On the left side, sec(45°) = √2, so sec^2(45°) = 2, and cosec(45°) = √2, so cosec^2(45°) = 2. The sum sec^2(45°) + cosec^2(45°) also equals 4. This confirms that the formula X = 2 + tan^2(A) + cot^2(A) is correct.

Why Other Options Are Wrong:
Option b gives 1 + tan^2(A) + cot^2(A), which would correspond to sec^2(A) + cosec^2(A) − 1 and therefore underestimates X by 1. Option c omits the constant 2 entirely. Option d, sec^2(A) * cosec^2(A), multiplies the squares of secant and cosecant and does not match the sum. Option e, tan^2(A) * cot^2(A), simplifies to 1, which is clearly different from X for most angles. Only option a matches the derived expression exactly.

Common Pitfalls:
Students sometimes confuse the identities sec^2(A) = 1 + tan^2(A) and cosec^2(A) = 1 + cot^2(A) with other formulas or forget the constant term 1. Another mistake is attempting to express sec(A) and cosec(A) directly in terms of sin(A) and cos(A) and then in terms of tan(A) and cot(A), which is longer and more error prone. Remembering the standard squared identities and applying them directly is the quickest and safest path to the correct answer.

Final Answer:
The expression X = sec^2(A) + cosec^2(A) can be written as 2 + tan^2(A) + cot^2(A).