Express the cosine of triple angle $3x$ in terms of cosines of $2x$ and $x$

Question

Show that $$\cos{3x}=2\cos{2x}\cos{x}-\cos{x}$$

I've tried adding and subtracting $\cos{x}$ from $\cos{3x}$, like this: $$\cos{3x}+\cos{x}-\cos{x}$$ so I get that $$\cos{3x}+\cos{x}=2\cos{2x}\cos{x}$$ But I have no idea how these are equal.

Answer 1

Hints:

$$\cos(x+y)=\cos x\cos y-\sin x\sin y$$

$$\sin2x=2\sin x\cos x$$

$$\cos{2x}=\cos^2{x}-\sin^2{x}=1-2\sin^2{x}\implies \sin^2{x}=\frac{1-\cos{2x}}2$$

With the above you're done.

Answer 2

Hint

$$\cos(3x)=\cos(2x+x)=\cos(2x)\cos(x)-\sin(2x)\sin(x)$$

and use the formula for $\sin(2x)$.

Answer 3

Use Prosthaphaeresis Formula, $\displaystyle\cos C+\cos D=\cdots$ on $\cos3x+cosx$

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Alternatively, $$\cos3x+cosx=\cos(2x+x)+\cos(2x-x)=?$$

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Alternatively use Werner Formula $$2\cos2x\cos x=\cos(2x+x)+cos(2x-x)$$