Using the compound–angle expansions for \(\cos(5x+x)\) and \(\cos(5x-x)\),
prove that
\[
\cos 6x+\cos 4x \;\equiv\; 2\cos 5x\cos x .
\]
Solution
Expand each cosine:
\[
\cos(5x+x)
= \cos 5x\cos x - \sin 5x\sin x,
\]
\[
\cos(5x-x)
= \cos 5x\cos x + \sin 5x\sin x.
\]
Add the two results:
\[
\cos 6x + \cos 4x
= (\cos 5x\cos x - \sin 5x\sin x)
+ (\cos 5x\cos x + \sin 5x\sin x)
= 2\cos 5x\cos x.
\]
Hence proved.
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