The Identities Calculus Assumes — Essentials

The one to know

$\sin^2 x + \cos^2 x = 1$

Sign trap when rearranging: $\sin^2 x - 1 = -\cos^2 x$, not $\cos^2 x$.

Definitions

$\tan x = \dfrac{\sin x}{\cos x}$, $\cot x = \dfrac{\cos x}{\sin x}$, $\sec x = \dfrac{1}{\cos x}$, $\csc x = \dfrac{1}{\sin x}$

Divided forms

$1 + \tan^2 x = \sec^2 x \qquad 1 + \cot^2 x = \csc^2 x$

Double angles

$\sin 2x = 2\sin x\cos x$

For integration

$\cos^2 x = \dfrac{1 + \cos 2x}{2} \qquad \sin^2 x = \dfrac{1 - \cos 2x}{2}$

Half a turn

$\cos(\pi + x) = -\cos x \qquad \sin(\pi + x) = -\sin x$

Stuck?

Rewrite everything in $\sin$ and $\cos$.