One can show by integrating the following function
$$f(z) = z^{n-m-1}(1+z^2)^m$$
around the following contour
By equating the circular part of $|z|=1$ and the line on the imaginary part.
$$\int^{\pi/2}_{0}\cos(nt)\cos^m(t)\,dt=2^{-m}\sin\left(\frac{n\pi -m\pi}{2} \right)\int^1_{0}t^{n-m-1}(1-t^2)^m\,dt \tag{1}$$
The left side looks close to the Beta representation
$$2\int^{\pi/2}_0 \sin^{2n-1}(t) \cos^{2m-1}(t)\,dt = \frac{\Gamma(n)\Gamma(m)}{\Gamma(n+m)} \tag{2}$$
Questions
Can we show (1) using elementary transformations ?
What real methods can we use to prove the integral
$$\int^{\pi/2}_{0}\cos(nt)\cos^m(t)\,dt=\frac{\pi \Gamma(m+1)}{2^{m+1}\Gamma\left(\frac{n+m+2}{2}\right)\Gamma\left(\frac{2-n+m}{2}\right)}\tag{3}$$