# Primitive of Reciprocal of Power of Cosine of a x by Power of Sine of a x/Reduction of Power of Cosine

## Theorem

$\displaystyle \int \frac {\mathrm d x} {\sin^m a x \cos^n a x} = \frac 1 {a \left({n - 1}\right) \sin^{m - 1} a x \cos^{n - 1} a x} + \frac {m + n - 2} {n - 1} \int \frac {\mathrm d x} {\sin^m a x \cos^{n - 2} a x}$

## Proof

 $\displaystyle \int \frac {\mathrm d x} {\sin^m a x \cos^n a x}$ $=$ $\displaystyle \int \frac {\sin^{-m} a x \ \mathrm d x} {\cos^n a x}$ $\displaystyle$ $=$ $\displaystyle \frac {\sin^{-m + 1} a x} {a \left({n - 1}\right) \cos^{n - 1} a x} - \frac {-m - n + 2} {n - 1} \int \frac {\sin^{-m} a x} {\cos^{n - 2} a x} \ \mathrm d x + C$ Primitive of $\dfrac {\sin^m a x} {\cos^n a x}$ $\displaystyle$ $=$ $\displaystyle \frac 1 {a \left({n - 1}\right) \sin^{m - 1} a x \cos^{n - 1} a x} + \frac {m + n - 2} {n - 1} \int \frac {\mathrm d x} {\sin^m a x \cos^{n - 2} a x} + C$

$\blacksquare$