4 ^ -1 1 5 - ^ -1 1 239 is equal to - Vidyadip

MCQ

$4 \tan ^{-1} \frac{1}{5}-\tan ^{-1} \frac{1}{239}$ is equal to

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

✓

Answer

Correct option: D.

$\frac{\pi}{4}$

(D) Since, $2 \tan ^{-1} x=\tan ^{-1} \frac{2 x}{1-x^2}$
$\therefore 4 \tan ^{-1} \frac{1}{5}=2\left[2 \tan ^{-1} \frac{1}{5}\right]=2 \tan ^{-1} \frac{\frac{2}{5}}{1-\frac{1}{25}}$
$=2 \tan ^{-1} \frac{10}{24}=\tan ^{-1} \frac{\frac{20}{24}}{1-\frac{100}{576}}=\tan ^{-1} \frac{120}{119}$
$\therefore 4 \tan ^{-1} \frac{1}{5}-\tan ^{-1} \frac{1}{239}=\tan ^{-1} \frac{120}{119}-\tan ^{-1} \frac{1}{239}$
$=\tan ^{-1} \frac{\frac{120}{119}-\frac{1}{239}}{1+\frac{120}{119} \cdot \frac{1}{239}}=\tan ^{-1} \frac{(120 \times 239)-119}{(119 \times 239)+120}$
$=\tan ^{-1} 1=\frac{\pi}{4}$

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