What is the value of [tan (90 – A) + cot (90 – A)]^2 / [2 sec^2(90 – 2A)] in trigonometry?

Introduction / Context:
This trigonometric question focuses on complementary angle identities. It uses basic relationships between tangent, cotangent, secant, and cosecant. The aim is to transform the expression step by step until it becomes a simple constant value.

Given Data / Assumptions:

• Expression E = [tan(90 - A) + cot(90 - A)]^2 / [2 sec^2(90 - 2A)].
• A is an angle for which all functions are defined.
• We must simplify E to a single numeric value.

Concept / Approach:
We use the complementary angle identities: tan(90 - A) = cot A and cot(90 - A) = tan A. For the secant term, we use sec(90 - θ) = csc θ. Then we express everything in terms of sin A and cos A and use the double angle identity for sine. This leads to major cancellations that produce a constant.

Step-by-Step Solution:
First, use tan(90 - A) = cot A and cot(90 - A) = tan A. Then tan(90 - A) + cot(90 - A) = cot A + tan A. So the numerator is (cot A + tan A)^2. Express cot A and tan A in terms of sine and cosine: tan A = sin A / cos A and cot A = cos A / sin A. Then cot A + tan A = cos A / sin A + sin A / cos A = (cos^2 A + sin^2 A) / (sin A cos A) = 1 / (sin A cos A). So the numerator becomes (1 / (sin A cos A))^2 = 1 / (sin^2 A cos^2 A). Now consider the denominator: 2 sec^2(90 - 2A). Using sec(90 - θ) = csc θ, we have sec(90 - 2A) = csc 2A. So sec^2(90 - 2A) = csc^2 2A = 1 / sin^2 2A. Therefore the denominator equals 2 / sin^2 2A. Now write E as E = [1 / (sin^2 A cos^2 A)] / [2 / sin^2 2A]. Use sin 2A = 2 sin A cos A, so sin^2 2A = 4 sin^2 A cos^2 A. Thus the denominator is 2 / (4 sin^2 A cos^2 A) = 1 / (2 sin^2 A cos^2 A). Hence E = [1 / (sin^2 A cos^2 A)] / [1 / (2 sin^2 A cos^2 A)] = 2.

Verification / Alternative check:
Take A = 30 degrees. Numerically evaluate both the original expression and 2 using a calculator. Both values agree, confirming that the simplification is correct.

Why Other Options Are Wrong:
1: This would result if the factor 2 in the denominator were handled incorrectly.
0 and -1: These would require cancellation that reduces everything to zero or a sign change, which does not occur here.
4: This would appear if the factor from sin 2A were doubled instead of halved, a common algebraic slip.

Common Pitfalls:
Common mistakes include forgetting that sec(90 - 2A) becomes csc 2A, mishandling the square on the entire bracket, or misusing the double angle identity for sine. Writing each identity clearly before substitution helps avoid confusion.

Final Answer:
The value of the given expression is 2.