Quadruple Angle Formulas

Theorem

$\sin 4 \theta = 4 \sin \theta \cos \theta - 8 \sin^3 \theta \cos \theta$

$\cos 4 \theta = 8 \cos^4 \theta - 8 \cos^2 \theta + 1$

$\tan 4 \theta = \dfrac {4 \tan \theta - 4 \tan^3 \theta} {1 - 6 \tan^2 \theta + \tan^4 \theta}$

where $\sin, \cos, \tan$ denote sine, cosine and tangent respectively.

$\sinh 4 x = 8 \sinh^3 x \cosh x + 4 \sinh x \cosh x$

$\cosh 4 x = 8 \cosh^4 x - 8 \cosh^2 x + 1$

$\tanh 4 x = \dfrac {4 \tanh x + 4 \tanh^3 x} {1 + 6 \tanh^2 x + \tanh^4 x}$

where $\sinh, \cosh, \tanh$ denote hyperbolic sine, hyperbolic cosine and hyperbolic tangent respectively.