Question

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

Answer 1

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

we know that

${\tan }^{-1}x+{\tan }^{-1}y=\{\begin{array}{c}{\tan }^{-1}\left(\frac{x+y}{1-xy}\right)ifxy<1\\ \pi +{\tan }^{-1}\left(\frac{x+y}{1+xy}\right)ifxy>1\end{array}$

Now, for $0<A<\frac{\pi }{4}\cot A>1$ and for $\frac{\pi }{4}<a<\frac{\pi }{2}\cot A<1$

$\therefore$ we can with

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

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

$[As\frac{\cot A+{\cot }^{3}A}{1-{\cot }^{4}A}=\frac{\cot A(1+{\cot }^{2}A)}{(1+{\cot }^{2}A)(1+{\cot }^{2}A)}]$

Now $\frac{\cot A}{1-{\cot }^{2}A}=\frac{\frac{1}{\tan \theta }}{1-\frac{1}{{\tan }^{2}A}}=\frac{\tan A}{{\tan }^{2}A-1}=\frac{1}{2}\tan 2A$

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

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

Adding ${\tan }^{-1}\left(\frac{1}{2}\tan 2A\right)$ or both sides we get

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