Question

The equation ${\tan }^{-1}x-{\cot }^{-1}x={\tan }^{-1}\left(\frac{1}{\sqrt{3}}\right)$ has

• A. no solution
• B. unique solution
• C. infinite number of solutions
• D. two solutions

Answer 1

The correct option is B unique solution

We have ${\tan }^{-1}x-{\cot }^{-1}x=\frac{\pi }{6}$
and we know that, ${\tan }^{-1}x+{\cot }^{-1}x=\frac{\pi }{2}$

Adding them, we get $2{\tan }^{-1}x=\frac{2\pi }{3}$

$\Rightarrow {\tan }^{-1}x=\frac{\pi }{3}$ i.e., $x=\sqrt{3}.$