$$L.H.S=\int\limits_0^{\frac \pi2}\tan^n x\space dx$$ $ =\int\limits_0^{\frac \pi2}\dfrac {\sin^n x}{\cos^n x}dx\\ =\int\limits_0^{\frac \pi2}\sin^nx.\cos^{-n}x dx =\dfrac 12 B\Bigg(\dfrac {n+1}2,\dfrac {-n+1}2\Bigg)\\ =\dfrac 12\dfrac {\Bigg)\overline{\dfrac {n+1}2}\Bigg)\overline{\dfrac {1-n}2}}{\Bigg)\overline{\dfrac {n+1+1-n}2}} \\ =\dfrac 12 \Bigg)\overline{\dfrac {n+1}2}\Bigg)\overline{\dfrac {1-n}2} \\ Let\space P=\dfrac {1-n}2\\ 1-p=1-\dfrac 12+\dfrac n2\\ =\dfrac {n+1}2\\ \therefore L.H.S=\dfrac 12\times \dfrac {\pi}{\sin \Bigg(\dfrac {1-n}2\Bigg)\pi}\\ Since \Bigg(\therefore )\overline{P})\overline{1-P}=\dfrac {\pi}{\sin P\pi}\Bigg)\\ L.H.S=\dfrac \pi2\times \dfrac 1{\sin \Bigg(\dfrac \pi2-\dfrac {n\pi}2\Bigg)}\\ =\dfrac \pi2\times \dfrac 1{\cos \dfrac {n\pi}2}\\ Since \space \sin(\dfrac \pi2-\theta)=\cos\theta\\ \therefore L.H.S=\dfrac {\pi}2\sec \dfrac {n\pi}2\\ =R.H.S$