Question

The value of $2{\cot }^{-1}\frac{1}{2}-{\cot }^{-1}\frac{4}{3}$ is

• A. $-\frac{\pi }{8}$
• B. $\frac{3\pi }{2}$
• C. $\frac{\pi }{4}$
• D. $\frac{\pi }{2}$

Answer 1

Answer: (D)

$2{\cot }^{-1}\frac{1}{2}-{\cot }^{-1}\frac{4}{3}$
$=2{\tan }^{-1}2-{\tan }^{-1}\frac{3}{4}$
$=\pi +{\tan }^{-1}\frac{4}{-3}-{\tan }^{-1}\frac{3}{4}$
$=\pi -{\tan }^{-1}\frac{4}{3}-{\tan }^{-1}\frac{3}{4}$
$=\pi -\left\{{\tan }^{-1}\frac{\frac{4}{3}+\frac{3}{4}}{1-\frac{4}{3}\cdot \frac{3}{4}}\right\}$
$=\pi -{\tan }^{-1}\infty$
$=\pi -\frac{\pi }{2}=\frac{\pi }{2}$