Q95 / 160 · WBJEE 2015 · Mathematics
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$\frac{\pi}{4}$
- Correct answerD$\frac{\pi}{2}$
Solution
$2 \cot ^{-1} \frac{1}{2}-\cot ^{-1} \frac{4}{3}$
$\begin{array}{l}
=2 \tan ^{-1} 2-\tan ^{-1} \frac{3}{4} \\
=\pi+\tan ^{-1} \frac{4}{-3}-\tan ^{-1} \frac{3}{4} \\
=\pi-\tan ^{-1} \frac{4}{3}-\tan ^{-1} \frac{3}{4}
\end{array}$
$=\pi-\left\{\tan ^{-1} \frac{\frac{4}{3}+\frac{3}{4}}{1-\frac{4}{3} \cdot \frac{3}{4}}\right\}$
$=\pi-\tan ^{-1} \infty=\pi-\frac{\pi}{2}=\frac{\pi}{2}$
Hence, the correct answer is option: 'D'.