If $n$ and $m$ are both even,
then the tricks of the previous slide do not help. Instead,
we can either integrate by parts (using the "go in a circle" trick
in the previous module) or use double-angle formulas. Most
people find the double-angle formulas to be easier, and that's
what this slide's video is about.

Remember the trig
identities we said would be helpful:
$$\sin^2(x) = \frac{1-\cos(2x)}{2}$$ $$\cos^2(x) =
\frac{1+\cos(2x)}{2}.$$This converts $\sin^n(x)\cos^m(x)$,
with $n$ and $m$ even, into terms of $\cos(2x)$. The odd
powers of $\cos(2x)$ can be handled as in the previous
slide. To integrate the even powers of $\cos(2x)$, apply
the double-angle trick again, getting a polynomial in
$\cos(4x)$. Repeat as many times as necessary. While
this can be tedious, it is not hard.