Question

The function cos(5θ) can be written as acos5(θ)−bcos3(θ)+ccos(θ) where a, b, and c are positive integers. Find a+b+c.

Answer 1

\[\cos(5x) = \cos(4x+x) = \cos(4x) \cos x - \sin(4x)\sin x\]
\[= \cos(3x+x)\cos x - \sin(3x+x) \sin x\]
\[=[\cos(3x)\cos x - \sin(3x)\sin x]\cos x - [\sin(3x) \cos x + \cos(3x)\sin x] \sin x\]
\[=\cos(2x+x)\cos^{2} x -2\sin(2x+x) \sin x \cos x - \cos(2x+x) \sin^{2} x\]
\[=\cos(2x+x) \cos(2x) - \sin(2x+x) \sin(2x)\]
\[=[\cos(2x) \cos x - \sin(2x)\sin x] \cos(2x) - [\sin(2x) \cos x + \cos(2x) \sin x]\sin(2x)\]
\[=\cos^{2}(2x) \cos x -2\sin(2x) \cos(2x) \sin x -\sin^{2}(2x) \cos x\]
\[=(2\cos^{2} x -1)^{2} \cos x - 2(2\sin x \cos x)(2\cos^{2}x-1)\sin x-(4\sin^{2} x \cos^{2}x)\cos x\]
\[=(4\cos^{5} x-4\cos^{3}x+\cos x)+(-8\sin^{2}x \cos^{3} x+4\sin^{2}x \cos x)-4\sin^{2} x \cos^{3} x\]
\[=4\cos^{5}x-4\cos^{3} x+\cos x-12\cos^{3}x(1-\cos^{2} x)+4\cos x(1-\cos^{2}x)\]
\[=16\cos^{5} x-20\cos^{3}x+5\cos x\]