Power Reduction Formulas

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Theorem

Square of Sine

$\sin^2 x = \dfrac {1 - \cos 2 x} 2$


Square of Cosine

$\cos^2 x = \dfrac {1 + \cos 2 x} 2$


Square of Tangent

$\tan^2x = \dfrac {1 - \cos2x} {1 + \cos2x}$


Cube of Sine

$\sin^3 x = \dfrac {3 \sin x - \sin 3 x} 4$


Cube of Cosine

$\cos^3 x = \dfrac {3 \cos x + \cos 3 x} 4$


Fourth Power of Sine

$\sin^4 x = \dfrac {3 - 4 \cos 2 x + \cos 4 x} 8$


Fourth Power of Cosine

$\cos^4 x = \dfrac {3 + 4 \cos 2 x + \cos 4 x} 8$


Fifth Power of Sine

$\sin^5 x = \dfrac {10 \sin x - 5 \sin 3 x + \sin 5 x} {16}$


Fifth Power of Cosine

$\cos^5 x = \dfrac {10 \cos x + 5 \cos 3 x + \cos 5 x} {16}$


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


Square of Hyperbolic Sine

$\sinh^2 x = \dfrac {\cosh 2 x - 1} 2$


Square of Hyperbolic Cosine

$\cosh^2 x = \dfrac {\cosh 2 x + 1} 2$


Cube of Hyperbolic Sine

$\sinh^3 x = \dfrac {\sinh 3 x - 3 \sinh x} 4$


Cube of Hyperbolic Cosine

$\cosh^3 x = \dfrac {\cosh 3 x + 3 \cosh x} 4$


Fourth Power of Hyperbolic Sine

$\sinh^4 x = \dfrac {3 - 4 \cosh 2 x + \cosh 4 x} 8$


Fourth Power of Hyperbolic Cosine

$\cosh^4 x = \dfrac {3 + 4 \cosh 2 x + \cosh 4 x} 8$


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


Comment

The identities for $\sin^m x$ and $\cos^n x$ can be useful for integrating expressions of the form:

$\displaystyle \int \sin^m x \ \cos^n x \ \mathrm d x$