Power Reduction Formulas

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

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

Sixth Power of Cosine

$\cos^6 x = \dfrac {10 + 15 \cos 2 x + 6 \cos 4 x + \cos 6 x} {32}$

Seventh Power of Cosine

$\cos^7 x = \dfrac {35 \cos x + 21 \cos 3 x + 7 \cos 5 x + \cos 7 x} {64}$

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:

$\ds \int \sin^m x \ \cos^n x \rd x$