Power Reduction Formulas

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Theorem

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

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

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

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

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle 1 - 2 \sin^2x$	$=$	$\displaystyle \cos2x$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Corollary to Double Angle Formulas
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \sin^2x$	$=$	$\displaystyle \frac {\cos2x - 1} {-2}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	solving for $\sin^2x$
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac {1 - \cos2x} 2$	$\displaystyle $	$\displaystyle $	$\displaystyle $	multiplying by $1 = \frac {-1}{-1}$ and rearranging terms

$\blacksquare$

	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle 2 \cos^2x - 1$	$=$	$\displaystyle \cos2x$	$\displaystyle $	$\displaystyle $	$\displaystyle $	Corollary to Double Angle Formulas
	$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \cos^2x$	$=$	$\displaystyle \frac {1 + \cos2x} 2$	$\displaystyle $	$\displaystyle $	$\displaystyle $	solving for $\cos^2x$

$\blacksquare$

$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle \tan^2x$	$=$	$\displaystyle \frac {\sin^2x} {\cos^2x}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	definition of tangent
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac {\frac {1 - \cos2x} 2} {\frac {\cos2x + 1} 2}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	from above
$\displaystyle $	$\displaystyle $	$\displaystyle $	$\displaystyle $	$=$	$\displaystyle \frac {1 - \cos2x} {1 + \cos2x}$	$\displaystyle $	$\displaystyle $	$\displaystyle $	multiplying by $1 = \frac 2 2$

$\blacksquare$

The identities for $\sin^2x$ and $\cos^2x$ can be useful for integrating expressions of the form:

$\displaystyle \int \sin^mx \ \cos^nx \ \mathrm dx$

where $m$ and $n$ are both even.

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle 1 - 2 \sin^2x\)	\(=\)	\(\displaystyle \cos2x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Corollary to Double Angle Formulas
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \sin^2x\)	\(=\)	\(\displaystyle \frac {\cos2x - 1} {-2}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	solving for $\sin^2x$
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {1 - \cos2x} 2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	multiplying by $1 = \frac {-1}{-1}$ and rearranging terms

	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle 2 \cos^2x - 1\)	\(=\)	\(\displaystyle \cos2x\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	Corollary to Double Angle Formulas
	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \cos^2x\)	\(=\)	\(\displaystyle \frac {1 + \cos2x} 2\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	solving for $\cos^2x$

\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \tan^2x\)	\(=\)	\(\displaystyle \frac {\sin^2x} {\cos^2x}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	definition of tangent
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {\frac {1 - \cos2x} 2} {\frac {\cos2x + 1} 2}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	from above
\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {1 - \cos2x} {1 + \cos2x}\)	\(\displaystyle \)	\(\displaystyle \)	\(\displaystyle \)	multiplying by $1 = \frac 2 2$