Chapter 10.3, Problem 26E

Show that the set of functions

$\left\{\cos \frac{\pi }{2}x,\sin \frac{\pi }{2}x,\cos \frac{3\pi }{2}x,\sin \frac{3\pi }{2}x,\cdots ,\cos \frac{\left(2n-1\right)\pi }{2}x,\sin \frac{\left(2n-1\right)\pi }{2}x,\mathrm{...}\right\}$

is an orthonormal system on $\left[-1,1\right]$ with respect to the weight function $w\left(x\right)\equiv 1$.