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

Theorem

 * $\ds \int \frac {\cos^m a x} {\sin^n a x} \rd x = \frac {-\cos^{m + 1} a x} {a \paren {n - 1} \sin^{n - 1} a x} - \frac {m - n + 2} {n - 1} \int \frac {\cos^m a x} {\sin^{n - 2} a x} \rd x + C$

Proof
Hence, rearranging:

Also see

 * Primitive of $\dfrac {\sin^m a x} {\cos^n a x}$ : Reduction of $\cos^n$