Question

The function cos(5θ) can be written as
a.cos^5(θ)−b.cos^3(θ)+c.cos^(θ) where a, b, and c are positive integers.
Find a+b+c.

Answer 1

\[\cos(5x) = \cos(3x+2x) = \cos(3x)\cos(2x)- \sin(3x) \sin(2x)\]
\[= \cos(2x+x)(2\cos^{2} x -1)- \sin(2x+x)\sin(2x)\]
\[=[\cos(2x)\cos x - \sin(2x) \sin x](2\cos^{2} x -1) - [\sin(2x)\cos x +\cos(2x) \sin x]\sin(2x)\]
\[=(2\cos^{2} x -1)^{2} \cos x - (2 \sin^{2}x \cos x)(2\cos^{2} x -1) - \sin^{2} (2x) \cos x - (2\sin^{2} x \cos x)\cos(2x)\]
\[=(4\cos^{5} x-4\cos^{3} x+\cos x)-(4\sin^{2} x \cos x)(2\cos^{2} -1) -(4\sin^{2} x \cos^{3} x)\]
\[=4\cos^{5} x-4\cos^{3} x+\cos x -12 \cos^{3} x (1-\cos^{2} x)+4\cos x(1-\cos^{2} x)\]
\[=16\cos^{5} x-20\cos^{3} x+5\cos x\]

Answer 2

hope you can follow that...its a lot of substitutions to get only cos