Question

prove that
cot^-1(1/5) +1/2 cot^-1(12/5)= pi/2
its urgent

Answer 1

Recall the identity $\tan^{-1}x+\tan^{-1}y=\pi+\tan^{-1}\frac{x+y}{1-xy}$ if $xy>1$

$\cot^{-1}(1/5)+1/2\cot^{-1}(12/5)\\~\\=\tan^{-1}(5)+1/2\tan^{-1}(5/12)\\~\\=\frac{1}{2}\left[2\tan^{-1}(5)+\tan^{-1}(5/12)\right]\\~\\=\frac{1}{2}\left[\tan^{-1}(5)+(\tan^{-1}(5)+\tan^{-1}(5/12))\right]\\~\\=\frac{1}{2}\left[\tan^{-1}(5)+(\pi+\tan^{-1}\frac{5+5/12}{1-5*5/12})\right]\\~\\=\frac{1}{2}\left[\tan^{-1}(5)+(\pi+\tan^{-1}(-5))\right]\\~\\=\frac{1}{2}\left[\tan^{-1}(5)+\pi-\tan^{-1}(5)\right]\\~\\=\frac{\pi}{2}$