I am working on proving or disproving $\cos^5(x)-\sin^5(x)=\cos(5x)$ True or false? $$\cos^5(x)-\sin^5(x)=\cos(5x)$$ for all real x.
I have no idea how to prove or disprove this. I tried to expand $\cos(5x)$ using double angle formula but I wasn't sure how to go from that to $$\cos^5(x)-\sin^5(x)$$

It is false. Quick counterexample: Note that, at $x=\frac{\pi}{5}$, we have $ \cos (5x) = -1 $, but clearly${}^\dagger$, $ \cos^5 \left( \frac {\pi}{5} \right) - \sin^5 \left( \frac {\pi}{5} \right) \ne 1 $. $\Box$

${}^\dagger$ It is clear that, since $ \frac {\pi}{5} $ is acute, $ 0 < \cos^5 \left( \frac {\pi}{5} \right), \sin^5 \left( \frac {\pi}{5} \right) < 1 $. Hence, $$ \cos^5 \left( \frac {\pi}{5} \right) - \sin^5 \left( \frac {\pi}{5} \right) < 1. $$This is how we get that it is not equal to $1$. 
