How do I express $\int_0^{\frac{\pi}{2}}\sin^{2m-1}\left(t\right)\cos^{2n-1}(t)dt$ using the Gamma-function?

Question

We define the Gamma function as: $$\Gamma(p)=\int_0^{+\infty}e^{-x}x^{p-1}dx$$

I was advised to rewrite the integral as $\sin(t)^{2m-1}\cos(t)^{2n-2} d \sin(t)$, and substitute $ t = \sin(t)$ which brings me to :

$$\int_0^1t^{2m-1}\left(1-t^2\right)^{n-1}dt$$

But I don't know how to go on from there.

Answer 1

Hint: $~e^{-x}=\lim\limits_{n\to\infty}\bigg(1-\dfrac xn\bigg)^n~:~$ Use this well-known identity, along with a simple

substitution, to bridge the gap between the integral expressions of the two functions.