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 $$\cos^5(x)-\sin^5(x)$$

$x=\dfrac{\pi}{4} \implies \cos^5 x-\sin^5 x=0 \neq\cos \dfrac{5\pi}{4}$.
