Question

${\tan }^{-1}\left(\frac{1}{2}\tan 2A\right)+{\tan }^{-1}\left(\cot A\right)+{\tan }^{-1}\left({\cot }^{3}A\right)=\{\begin{array}{c}0,if\frac{\pi }{4}<A<\frac{\pi }{2}\\ \pi ,if0<A<\frac{\pi }{2}\end{array}$

Answer 1

${\tan }^{-1}\left(\frac{1}{2}\tan 2A\right)+{\tan }^{-1}\left(\frac{\cot A+\cot 3A}{1-\cot 4A}\right)$

${\tan }^{-1}\left(\frac{2\tan A}{2(1-{\tan }^{2}x)}\right)+{\tan }^1\left(\frac{\cot A}{1-{\cot }^2}\right)$

$\Rightarrow {\tan }^{-1}\left(\frac{\tan A}{1-{\tan }^{2}A}\right)+{\tan }^{-1}\left(\frac{\frac{1}{\tan A}}{\frac{{\tan }^{2}A-1}{{\tan }^{2}A}}\right)$

$\Rightarrow {\tan }^{-1}\left(\frac{\tan A}{1-{\tan }^{2}A}\right)+{\tan }^{-1}\left(\frac{\tan A}{{\tan }^{2}A-1}\right)$

$\Rightarrow$ if $\frac{\pi }{4}<A<\frac{\pi }{2}$

$\Rightarrow {\tan }^{-1}\left(\frac{\tan A}{1-{\tan }^{2}A}\right)-{\tan }^{-1}\left(\frac{\tan A}{1-{\tan }^{2}A}\right)=0$

if $0<A<\frac{\pi }{2}$

$\Rightarrow {\tan }^{-1}\left(\frac{\tan A}{1-{\tan }^{2}A}\right)+{\cot }^{-1}\left(\frac{\tan A}{1-{\tan }^{2}A}\right)$

$=\pi$