Chapter 5.5, Problem 100E

(a) If f is continuous, prove that

${\displaystyle {\int }_0^{\pi /2}f(\cos x)dx={\displaystyle {\int }_0^{\pi /2}f(\sin x)dx}}$

(b) Use part (a) to evaluate ${\displaystyle {\int }_0^{\pi /2}{\cos }^{2}xdx}$ and ${\displaystyle {\int }_0^{\pi /2}{\sin }^{2}xdx}$.