Beta Function of Half with Half/Proof 2

Proof
By definition of the Beta function:


 * $\ds \map \Beta {x, y} := \int_{\mathop \to 0}^{\mathop \to 1} t^{x - 1} \paren {1 - t}^{y - 1} \rd t$

Thus:

Let $t = \sin^2 \theta$.

Then:
 * $\rd t = 2 \sin \theta \cos \theta \rd \theta$

and:

and so: