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  4. The value of 2 cot -1 (1/2)- cot -1 (4/3) is

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

WBJEEWBJEE 2015Inverse Trigonometric Functions

Solution:

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