2021/110) express $\cos\, 5t$ in term of power of $\cos\, t$

We have

$\cos\, 5t = (\cos\, 5 t + \cos\, t) - \cos\, t$

$= 2 \cos\, 3t \cos\, 2t - \cos\, t$ using $\cos\, A + \cos\, B = 2 \cos \frac{A+B}{2} \cos \frac{A-B}{2}$

$= 2 * (4 \cos ^3 t - 3 \cos\, t)(2\cos ^2 t - 1) - \cos\, t$ using formula for $cos 3t$ and $cos 2t$

$= 16 \cos^5 t - 20 \cos^3 t + 6 \cos t -\cos t$

$= 16 \cos^5 t - 20 \cos^3 t + 5 \cos t$