How do I calculate the Beta-function $B(m,n) = 2\int_0^{\frac{\pi}{2}}\sin ^{2 m-1}(t) \cos ^{2 n-1}(t)\, dt$

Question

The Beta-Function $$B(m,n) =2\int_0^{\frac{\pi}{2}}\sin ^{2 m-1}(t) \cos ^{2 n-1}(t)\, dt \tag{a}$$

is equal to

$$\frac{n-1}{m}B(m-1,n+1) \tag{b}.$$

How do I go from (a) to (b)? (I tried with partial integration but it gets hairy.)

(We suppose that m and n are positive integers)

Answer 1

Hint: See Wallis' integrals. All you need is a trigonometric substitution. $($ I assume you are already familiar with the integral expression for the beta function $)$.