In trigonometry, the triple-angle identity rewrites cos(3A) using only cos(A).\nWhich of the following expressions is the correct identity for cos 3A in terms of cos A?

Introduction / Context:
This question tests a standard trigonometric identity: the triple-angle formula for cosine. Triple-angle identities are used to simplify expressions, solve trigonometric equations, and convert higher-angle terms like cos(3A) into powers of cos(A).

Given Data / Assumptions:

• We need an identity for cos(3A) written only in terms of cos(A).
• A is any real angle (identity must hold for all A).

Concept / Approach:
The known triple-angle identity for cosine is derived from angle addition: cos(3A) = cos(2A + A) and then applying cos(2A) and sin(2A) formulas. The final form becomes a cubic polynomial in cos(A).

Step-by-Step Solution:
Start with cos(3A) = cos(2A + A) Use cos(u+v) = cos u * cos v - sin u * sin v So cos(3A) = cos(2A)*cos(A) - sin(2A)*sin(A) Replace cos(2A) = 2cos^2(A) - 1 and sin(2A) = 2sin(A)cos(A) Then cos(3A) = (2cos^2(A) - 1)cos(A) - (2sin(A)cos(A))sin(A) = 2cos^3(A) - cos(A) - 2sin^2(A)cos(A) Use sin^2(A) = 1 - cos^2(A) = 2cos^3(A) - cos(A) - 2(1 - cos^2(A))cos(A) = 2cos^3(A) - cos(A) - 2cos(A) + 2cos^3(A) = 4cos^3(A) - 3cos(A)

Verification / Alternative check:
Check A = 0: LHS cos(0) = 1. RHS 4*1 - 3*1 = 1, so it matches. This quick check supports the identity choice.

Why Other Options Are Wrong:
3 cos A - 4 cos^3 A is just the negative of the correct expression.
4 cos^3 A + 3 cos A has the wrong sign on the linear term and fails at A = 0.
cos A + 4 cos^3 A and 4 cos A - 3 cos^3 A do not match the known cubic structure and fail simple substitutions.

Common Pitfalls:
Mixing up the signs, or confusing the cosine triple-angle identity with the sine triple-angle identity (sin 3A). Another common error is writing cos(3A) as 3cos(A) - 4cos^3(A) without noticing it is the negative.

Final Answer:
cos(3A) = 4 cos^3 A - 3 cos A