Question

The value of ${\tan }^{-1}\left(\frac{1}{2}\tan 2A\right)+{\tan }^{-1}\left(\cot A\right)+{\tan }^{-1}\left({\cot }^{3}A\right)$ for $0<A<\frac{\pi }{4}$ is?

• A. $4{\tan }^{-1}1$
• B. $2{\tan }^{-1}2$
• C. $0$
• D. $None$

Answer 1

The correct option is A $4{\tan }^{-1}1$

We have,

$\begin{array}{c}{\tan }^{-1}\left(\frac{1}{2}\tan 2A\right)+{\tan }^{-1}\left(\cot A\right)+{\tan }^{-1}\left(\cot 3A\right)\\ {\tan }^{-1}\left(\frac{1}{2}\tan 2A\right)+\pi +{\tan }^{-1}\left(\frac{\cot A+{\cot }^{3}A}{1-{\cot }^{4}A}\right)\\ \tan \left(\frac{\tan A}{1-\tan A}\right)+\pi +{\tan }^{-1}\left(\frac{\cot A}{1-{\cot }^{2}A}\right)\\ =\pi +{\tan }^{-1}\left(\frac{\tan A}{{\tan }^{2}A-1^2}\right)+{\tan }^{-1}\left(\frac{\tan A}{1-{\tan }^{2}A}\right)\\ =\pi \\ =4{\tan }^{-1}\left(1\right)\\ Hence,optionAisthecorrectanswer.\end{array}$