# Primitive of Power of Sine of a x over Power of Cosine of a x/Reduction of Power of Cosine

## Theorem

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

## Proof

 $\displaystyle$  $\displaystyle \frac {\sin^{m + 1} a x} {a \left({n - 1}\right) \cos^{n - 1} a x} - \frac {m + 1} {n - 1} \int \frac {\sin^m a x} {\cos^{n - 2} a x} \ \mathrm d x + C$ $\displaystyle$ $=$ $\displaystyle \int \frac {\sin^{m + 2} a x} {\cos^n a x} \ \mathrm d x$ Primitive of $\dfrac {\sin^m a x} {\cos^n a x}$: Reduction of Both Powers $\displaystyle$ $=$ $\displaystyle \int \frac {\sin^m a x \left({1 - \cos^2 a x}\right)} {\cos^n a x} \ \mathrm d x$ Sum of Squares of Sine and Cosine $\displaystyle$ $=$ $\displaystyle \int \frac {\sin^m a x} {\cos^n a x} \ \mathrm d x - \int \frac {\sin^m a x \cos^2 a x} {\cos^n a x} \ \mathrm d x$ Linear Combination of Integrals $\displaystyle$ $=$ $\displaystyle \int \frac {\sin^m a x} {\cos^n a x} \ \mathrm d x - \int \frac {\sin^m a x} {\cos^{n - 2} a x} \ \mathrm d x$ simplifying

Hence, rearranging:

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

$\blacksquare$