Solution
= Solution
For every integer $m$, <De Moivre's theorem> states $(\cos\theta+i\sin\theta)^m=\cos(m\theta)+i\sin(m\theta)$. Taking the real part for $m=5$ gives
$$
\cos5\theta=\cos^5\theta-10\cos^3\theta\sin^2\theta+5\cos\theta\sin^4\theta.
$$
Replace $\sin^2\theta$ by $1-\cos^2\theta$ and expand:
$$
\boxed{\cos5\theta=16\cos^5\theta-20\cos^3\theta+5\cos\theta.}
$$
The resulting polynomial is the fifth <Chebyshev polynomial>.