$I=\int\limits_0^{\dfrac{\pi}2}\dfrac {\sin^n x}{\cos^n x}dx \hspace{3 cm} since \tan x=\dfrac {\sin x}{\cos x}$ $=\int\limits_0^{\dfrac{\pi}2}\sin^n x \cos x \text{dx}\hspace{3cm } $ since limits are also satisfied Now by beta gamma function. This can be written as $=\dfrac 12 B\Bigg(\dfrac {n+1}2,\dfrac {-n+1}2\Bigg)$ $=\dfrac 12 \dfrac {\sqrt{\dfrac {(n+1)}2}.\sqrt{\dfrac {(-n+1)}2}}{\sqrt1} ----- (1)$ Now we have …