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Simplify the trigonometric expression: (1 - sin(90° - 2A)) / (1 + sin(90° + 2A)).

Difficulty: Medium

Correct Answer: tan^2 A

Explanation:


Introduction / Context:
This question tests co-function identities involving 90 degrees and then the classic algebraic simplification of (1 - cos x) / (1 + cos x). After rewriting the sine terms, the expression collapses into a perfect square ratio of sine and cosine, giving tan^2 A.


Given Data / Assumptions:

  • Expression: (1 - sin(90° - 2A)) / (1 + sin(90° + 2A))
  • Use identities:
    • sin(90° - x) = cos x
    • sin(90° + x) = cos x
    • 1 - cos 2A = 2*sin^2 A
    • 1 + cos 2A = 2*cos^2 A


Concept / Approach:
Convert both sine terms into cos(2A). Then use double-angle identities to convert the numerator and denominator into squares of sin A and cos A and simplify the ratio.


Step-by-Step Solution:

Step 1: sin(90° - 2A) = cos(2A). Step 2: sin(90° + 2A) = cos(2A). Step 3: Expression becomes (1 - cos(2A)) / (1 + cos(2A)). Step 4: Use identities: 1 - cos(2A) = 2*sin^2 A and 1 + cos(2A) = 2*cos^2 A. Step 5: So the ratio is (2*sin^2 A) / (2*cos^2 A) = sin^2 A / cos^2 A. Step 6: sin^2 A / cos^2 A = (sin A / cos A)^2 = tan^2 A.


Verification / Alternative check:
Try A = 45 degrees: cos(90) = 0, so expression = (1 - 0)/(1 + 0) = 1, and tan^2 45 degrees = 1. Matches perfectly.


Why Other Options Are Wrong:

cot 2A or tan 2A: wrong angle and not squared. sin A * cos A: product form, not a ratio. cos^2 A: would require numerator to be cos^2 and denominator 1.


Common Pitfalls:
Incorrectly thinking sin(90° + x) becomes -cos x (it is +cos x), or forgetting the double-angle conversions.


Final Answer:
tan^2 A

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