[Math] Easy ways to remember trigonometric identities

Question

Are there any easy ways or mnemonics to memorize the trigonometric identities like for example
$$
\sin(3x) = 3\sin(x) – 4\sin^3(x)
$$
I find them quite difficult to come up with, I almost always need to look them up.

Answer 1

For that kind of trigonometric identity, you want to remember and use de Moivre's formula: $$ (\cos x + i \sin x)^n = \cos (nx) + i \sin (nx) $$ You need to be comfortable with complex numbers, though.

For the example you gave, you get $$ (\cos x + i \sin x)^3 = \cos^3 x + 3i \cos^2 x \sin x - 3 \cos x \sin^2 x - i\sin^3 x $$ and so, looking at the imaginary part, $$ \begin{eqnarray} \sin 3x &=& 3 \cos^2 x \sin x - \sin^3 x \\ &=& 3(1-\sin^2 x)\sin x- \sin^3 x \\ &=& 3\sin x -4\sin^3 x \end{eqnarray} $$