Addition and double-angle formulae

The idea

$$\sin(A \pm B) = \sin A\cos B \pm \cos A\sin B$$$$\cos(A \pm B) = \cos A\cos B \mp \sin A\sin B$$With $A = B$: $\sin 2A = 2\sin A\cos A$, and $\cos 2A$ can be written $\cos^2 A - \sin^2 A$, $2\cos^2 A - 1$ or $1 - 2\sin^2 A$.

1. Split an angle into ones you know: $75^\circ = 45^\circ + 30^\circ$.
2. Pick the form of $\cos 2A$ that leaves one trig function.
3. Simplify and solve or prove.

Watch out $\sin 2A$ is $2\sin A\cos A$, not $2\sin A$.