4 ^ -1 1 5 - ^ -1 1 70 + ^ -1 1 99 = - Vidyadip

MCQ

$4 \tan ^{-1} \frac{1}{5}-\tan ^{-1} \frac{1}{70}+\tan ^{-1} \frac{1}{99}=$

A
$\frac{\pi}{2}$
B
$\frac{\pi}{3}$
✓
$\frac{\pi}{4}$
D
$\frac{\pi}{6}$

✓

Answer

Correct option: C.

$\frac{\pi}{4}$

(C) $4 \tan ^{-1} \frac{1}{5}-\tan ^{-1} \frac{1}{70}+\tan ^{-1} \frac{1}{99}$
$=\tan ^{-1} \frac{120}{119}+\tan ^{-1} \frac{1}{99}-\tan ^{-1} \frac{1}{70}$
$=\tan ^{-1}\left(\frac{120}{119}\right)+\tan ^{-1}\left[\frac{\frac{1}{99}-\frac{1}{70}}{1+\frac{1}{99} \cdot \frac{1}{70}}\right]$
$=\tan ^{-1}\left(\frac{120}{119}\right)+\tan ^{-1}\left(\frac{-29}{6931}\right)$
$=\tan ^{-1} \frac{120}{119}-\tan ^{-1} \frac{29}{6931}$
$=\tan ^{-1} \frac{120}{119}-\tan ^{-1} \frac{1}{239}$
$=\tan ^{-1} 1=\frac{\pi}{4}$

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