Reduction Formula for Integral of Power of Sine/Proof 2

Theorem
Let $n \in \Z_{> 0}$ be a (strictly) positive integer.

Then:
 * $\displaystyle \int \sin^n x \ \mathrm d x = \dfrac {n - 1} n \int \sin^{n - 2} x \ \mathrm d x - \dfrac {\sin^{n-1} x \cos x} n$

is a reduction formula for $\displaystyle \int \sin^n x \ \mathrm d x$.

Proof
With a view to expressing the problem 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:

and let:

Then: