Reduction Formula for Integral of Power of Tangent

Theorem
For all $n \in \Z_{> 1}$:

Let:
 * $I_n := \ds \int \tan^n x \rd x$

Then:
 * $I_n = \dfrac {\tan^{n - 1} x} {n - 1} - I_{n - 2}$

is a reduction formula for $\ds \int \tan^n x \rd x$.

Proof
Let:

Then:

Hence the result.

Also see

 * Reduction Formula for Integral of Power of Sine
 * Reduction Formula for Integral of Power of Cosine