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

Theorem

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

Proof
With a view to expressing the primitive in the form:
 * $\displaystyle \int u \frac {\mathrm d v}{\mathrm d x} \ \mathrm d x = u v - \int v \frac {\mathrm d u}{\mathrm d x} \ \mathrm d x$

let:

Thus:

and so:

Then let:

Then: