<s>[INST] Prove: $\cos^3 \frac x3 + \cos^3 \frac{x+2\pi}{3} + \cos^3 \frac{x+4\pi}{3} = \frac{3}{4} \cos x$ I tried to solve the following identity:
$$\cos^3 \frac x3 + \cos^3 \frac{x+2\pi}{3} + \cos^3 \frac{x+4\pi}{3} = \frac{3}{4} \cos x$$ I applied formulas, $\cos (a+b)$ and $\cos^3 x$ and I arrived at
$$\cos 3x + 2 \cos x - 3 \cos \frac{x}{3}=0 $$
After this I am stuck and any formula I applied, got complicated. Can someone give me an idea as to how I could prove this identity?
 [/INST] The formula you arrived at is wrong (put $x=\frac {\pi}{2}$ to see that). A correct method would be this:
$$\cos 3x=4\cos^3 x-3\cos x$$
So, $$\cos^3 x=\frac {\cos 3x+3\cos x}{4}$$
Now you just have to apply this formula to the given three angles.
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