Difficulty: Medium
Correct Answer: cos^3 A - 3 sin^2 A cos A
Explanation:
Introduction / Context:
This question tests knowledge of multiple-angle identities, specifically the triple-angle identity for cosine. While cos 3A is often memorized as 4cos^3 A - 3cos A, it can also be expressed in mixed form involving both sin A and cos A. Understanding how these forms connect using sin^2 A + cos^2 A = 1 helps avoid confusion and makes it easier to recognize equivalent identities in different-looking algebraic formats.
Given Data / Assumptions:
Concept / Approach:
Start from the known triple-angle identity:
cos 3A = 4cos^3 A - 3cos A.
Rewrite the -3cos A part using sin^2 A + cos^2 A = 1:
-3cos A = -3cos A(sin^2 A + cos^2 A) = -3sin^2 A cos A - 3cos^3 A.
Then:
4cos^3 A - 3cos A = 4cos^3 A - 3cos^3 A - 3sin^2 A cos A
= cos^3 A - 3sin^2 A cos A.
That matches the required form.
Step-by-Step Solution:
1) Use the standard identity:
cos 3A = 4cos^3 A - 3cos A
2) Insert 1 = sin^2 A + cos^2 A into the -3cos A term:
-3cos A = -3cos A(sin^2 A + cos^2 A)
3) Expand:
-3cos A = -3sin^2 A cos A - 3cos^3 A
4) Substitute back:
cos 3A = 4cos^3 A + (-3sin^2 A cos A - 3cos^3 A)
5) Combine like terms:
cos 3A = (4cos^3 A - 3cos^3 A) - 3sin^2 A cos A
cos 3A = cos^3 A - 3sin^2 A cos A
Verification / Alternative check:
Test A = 0°:
cos 3A = cos 0° = 1.
Right side: cos^3 0° - 3sin^2 0° cos 0° = 1^3 - 3*0*1 = 1. Works.
Test A = 60°:
cos 180° = -1.
cos 60° = 1/2, sin 60° = √3/2.
Compute: (1/2)^3 - 3*(3/4)*(1/2) = 1/8 - 9/8 = -1. Works.
Why Other Options Are Wrong:
• Options with +3sin^2 A cos A or ±4sin^2 A cos A have incorrect coefficients/signs compared to the derived identity.
• 3cos A - 4cos^3 A equals -(4cos^3 A - 3cos A) = -cos 3A, so it has the wrong sign.
Common Pitfalls:
• Remembering 4cos^3 A - 3cos A but failing to convert it correctly into a sin^2 and cos form.
• Mixing the triple-angle identity for sine with the one for cosine.
Final Answer:
cos^3 A - 3 sin^2 A cos A
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