Trig Formulas That Win

$\sin(a\pm b)=\sin a\cos b\pm\cos a\sin b$; $\cos(a\pm b)=\cos a\cos b\mp\sin a\sin b$; $\tan(a+b)=\dfrac{\tan a+\tan b}{1-\tan a\tan b}$; $\sin 2a = 2\sin a\cos a$, $\cos 2a = 1-2\sin^2 a$.

In plain terms. Every other identity is a rearrangement of these. When an equation mixes angles, rewrite everything in terms of ONE angle with the addition formulas.

Example. $\tan a=\tfrac12$, $\tan b=\tfrac13$: $\tan(a+b) = \dfrac{\tfrac12+\tfrac13}{1-\tfrac16} = 1$, so $a+b = 45^\circ$.

$\cos 3\theta = 4\cos^3\theta - 3\cos\theta$ and $\sin 3\theta = 3\sin\theta - 4\sin^3\theta$. Cubic expressions in $\sin$ or $\cos$ are usually a triple angle wearing a costume.

In plain terms. See $8\cos^3\theta - 6\cos\theta$? That is $2\cos 3\theta$. Solving the cubic head-on is the trap; recognizing the pattern is the whole problem.

Example. $8\cos^3\theta-6\cos\theta=1 \Rightarrow \cos 3\theta=\tfrac12 \Rightarrow \theta=20^\circ$ (acute).

$\sin\theta\,\sin(60^\circ-\theta)\,\sin(60^\circ+\theta)=\dfrac{\sin 3\theta}{4}$, and the same shape holds for cosine and tangent ($\tan$ version multiplies to $\tan 3\theta$).

In plain terms. Three angles in arithmetic progression around 60° multiply into one triple angle — the engine behind products like $\sin10^\circ\sin50^\circ\sin70^\circ = \tfrac18$.

Example. $\tan20^\circ\tan40^\circ\tan80^\circ = \tan 60^\circ = \sqrt3$.