 Trigonometric form of the beta function

For positive real $p,q$, substitute $x={\sin }^2\theta$ in the defining Euler beta function integral. The increasing substitution maps $[0,\pi /2]$ onto $[0,1]$ and has $dx=2\sin \theta \cos \theta d\theta$. Combining this factor with $x^{p-1}(1-x{)}^{q-1}$ gives the displayed identity. Positivity of both parameters ensures convergence at the two endpoints, even when the integrand is unbounded there.