Question

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

• A. $0;if\pi /4<A<\pi /2$
• B. $0;if0<A<\pi /4$
• C. $\pi ;if\pi /4<A<\pi /2$
• D. $\pi ;if0<A<\pi /2$

Answer 1

Answer: (A), (D)

The correct options are
A $0;if\pi /4<A<\pi /2$
D $\pi ;if0<A<\pi /2$

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 $0<A<\frac{\pi }{4}$ we have $\cot A>1$

and for $\frac{\pi }{4}<A<\frac{\pi }{2}$ we have $\cot A<1$

$\therefore$ we can write

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

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

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

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

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

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

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

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

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