Chapter 5.9, Problem 55ES

Given that $n$ is a positive integers, show that

${\displaystyle {\int }_0^{\pi /2}{\sin }^{n}xdx}={\displaystyle {\int }_0^{\pi /2}{\cos }^{n}xdx}$

by using a trigonometric identity and making a substitution. Do not attempt to evaluate the integrals.