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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.

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    In addition to the other responses, it is fairly easy to derive them using the sum-difference trig formulas, so remembering those can be used to derive all other trig identities.2012-10-18

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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} $

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In general, you can manipulate the following definitions of $\sin$ and $\cos$ in order to derive many identities:

$e^{i\theta} = \cos \theta + i\sin \theta$

$\sin \theta := \frac{e^{i\theta}-e^{-i\theta}}{2 i}$

$\cos \theta := \frac{e^{i\theta}+e^{-i\theta}}{2}$

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    Look here http://math.stackexchange.com/questions/193387/chebyshev-polynomials2015-11-30