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Simplify and evaluate: sin(90° + 2A) * [4 - cos^2(90° - 2A)].

Difficulty: Hard

Correct Answer: 4*(cos^6 A - sin^6 A)

Explanation:


Introduction / Context:
This question tests co-function identities, power manipulations, and recognizing higher-power patterns like cos^6 A - sin^6 A. The notation cos^2(90° - 2A) means the square of cos(90° - 2A). After converting the 90-degree shifts, the expression becomes a product in sin A and cos A that matches the factorization of cos^6 A - sin^6 A.


Given Data / Assumptions:

  • Expression: sin(90° + 2A) * [4 - cos^2(90° - 2A)]
  • Identities:
    • sin(90° + x) = cos x
    • cos(90° - x) = sin x
    • cos 2A = cos^2 A - sin^2 A
    • sin 2A = 2*sin A*cos A


Concept / Approach:
Convert the shifted angles to basic sin and cos of 2A, rewrite everything in terms of sin A and cos A, and match it to 4*(cos^6 A - sin^6 A) using factorization: cos^6 - sin^6 = (cos^2 - sin^2)(cos^4 + cos^2 sin^2 + sin^4).


Step-by-Step Solution:

Step 1: sin(90° + 2A) = cos 2A. Step 2: cos(90° - 2A) = sin 2A, so cos^2(90° - 2A) = sin^2 2A. Step 3: Expression becomes cos 2A * [4 - sin^2 2A]. Step 4: Write sin^2 2A = (2*sin A*cos A)^2 = 4*sin^2 A*cos^2 A. Step 5: So [4 - sin^2 2A] = 4 - 4*sin^2 A*cos^2 A = 4*(1 - sin^2 A*cos^2 A). Step 6: Also cos 2A = cos^2 A - sin^2 A. Step 7: Expression = 4*(cos^2 A - sin^2 A)*(1 - sin^2 A*cos^2 A). Step 8: Note that (1 - sin^2 A*cos^2 A) = cos^4 A + cos^2 A*sin^2 A + sin^4 A (since (cos^2 + sin^2)^2 = 1 gives cos^4 + 2cos^2 sin^2 + sin^4 = 1, rearrange to get cos^4 + sin^4 = 1 - 2cos^2 sin^2, then add cos^2 sin^2 to get 1 - cos^2 sin^2). Step 9: Therefore expression = 4*(cos^2 A - sin^2 A)*(cos^4 A + cos^2 A*sin^2 A + sin^4 A) = 4*(cos^6 A - sin^6 A).


Verification / Alternative check:
Take A = 45°: cos^6 A - sin^6 A = 0, so RHS = 0. LHS: cos 90° = 0, so LHS = 0. Matches.


Why Other Options Are Wrong:

4*(cos^6 A + sin^6 A): uses a plus, but the factor (cos^2 - sin^2) creates a difference. 2*(cos^3 A ± sin^3 A): wrong power structure. cos 2A: misses the large multiplier and squared term effect.


Common Pitfalls:
Misreading cos^2(90° - 2A) as cos(2*(90° - 2A)) instead of (cos(90° - 2A))^2, and failing to convert sin 2A correctly.


Final Answer:
4*(cos^6 A - sin^6 A)

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