Đặt $x=\dfrac{\pi}{2} -t \Rightarrow dx=-dt$
$I=\int \limits_{0}^{\frac{\pi}{2}} \dfrac{\cos^{20} t}{\sin^{20} t + \cos^{20} t }dt =\int \limits_{0}^{\frac{\pi}{2}} \dfrac{\cos^{20} x}{\sin^{20} x + \cos^{20} x }dx$
$\Rightarrow I + I = \int \limits_{0}^{\frac{\pi}{2}} \dfrac{\sin^{20} x}{\sin^{20} x + \cos^{20} x }dx+\int \limits_{0}^{\frac{\pi}{2}} \dfrac{\cos^{20} x}{\sin^{20} x + \cos^{20} x }dx =\int dx = x \bigg|_0^{\frac{\pi}{2}}$
$=\dfrac{\pi}{2} \Rightarrow I=\dfrac{\pi}{4}$