📦 Box of usefulness

Box of usefulness

$\tan^2 x + 1 \equiv \sec^2 x$
$1 + \cot^2 x \equiv \cosec^2 x$
$\sin 2x \equiv 2\sin x\cos x$
$\cos 2x \equiv 2\cos^2 x - 1$
$\cos 2x \equiv 1 - 2\sin^2 x$

Still useful!

$\sin^2 x + \cos^2 x \equiv 1$
$\cos 2x \equiv \cos^2 x - \sin^2 x$
$\tan 2x \equiv \frac{2\tan x}{1 - \tan^2 x}$

Adapting the box of usefulness

They work for any angle: $\cos 6x \equiv 1 - 2\sin^2 3x$, $\ \sin 14x \equiv 2\sin 7x\cos 7x$

Addition formulae (for products)

$\cos(A \pm B) \equiv \cos A\cos B \mp \sin A\sin B$
$\sin(A \pm B) \equiv \sin A\cos B \pm \cos A\sin B$

Integrals you already know

$\int \cos kx\,dx = \frac{1}{k}\sin kx + c$, $\ \int \sin kx\,dx = -\frac{1}{k}\cos kx + c$
$\int \sec^2 kx\,dx = \frac{1}{k}\tan kx + c$, $\ \int \cosec^2 kx\,dx = -\frac{1}{k}\cot kx + c$
$\int \sec kx\tan kx\,dx = \frac{1}{k}\sec kx + c$, $\ \int \cosec kx\cot kx\,dx = -\frac{1}{k}\cosec kx + c$
$\int \tan kx\,dx = \frac{1}{k}\ln|\sec kx| + c$, $\ \int \cot kx\,dx = \frac{1}{k}\ln|\sin kx| + c$

Strategy: use trig identities and manipulation to rewrite the integrand as terms you can integrate. Very important: as you are doing so much manipulation to get to something easier to integrate, you can often forget to integrate!