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STD 12 - 2. inverse trigonometric function — Mathematics STD 12 Science — Question

CBSE BoardEnglish MediumSTD 12 ScienceMathematicsSTD 12 - 2. inverse trigonometric function1 Mark
MCQ
$4tan^{-1} \frac{1}{5} -tan^{-1} \frac{1}{239}$ is equal to
  • A
    $\pi$
  • B
    $\frac{\pi}{2}$
  • C
    $\frac{\pi}{3}$
  • ✓
    $\frac{\pi}{4}$
✓

Answer

Correct option: D.
$\frac{\pi}{4}$
d
$4 \tan ^{-1} \frac{1}{5}-\tan ^{-1} \frac{1}{239}$

$=2\left(2 \tan ^{-1} \frac{1}{5}\right)-\tan ^{-1} \frac{1}{239}$

$=2 \tan ^{-1} \frac{\frac{2}{5}}{1-\left(\frac{1}{5}\right)^{2}}-\tan ^{-1} \frac{1}{239} \quad\left(\because 2 \tan ^{-1} x=\tan ^{-1} \frac{2 x}{1-x^{2}}\right)$

$=2 \tan ^{-1} \frac{2 / 5}{24 / 25}-\tan ^{-1} \frac{1}{239}$

$=2 \tan ^{-1} \frac{5}{12}-\tan ^{-1} \frac{1}{239}$

$=2 \tan ^{-1} \frac{\frac{2}{5}}{1-\left(\frac{1}{5}\right)^{2}}-\tan ^{-1} \frac{1}{239} \quad\left(\because 2 \tan ^{-1} x=\tan ^{-1} \frac{2 x}{1-x^{2}}\right)$

$=2 \tan ^{-1} \frac{2 / 5}{24 / 25}-\tan ^{-1} \frac{1}{239}$

$=2 \tan ^{-1} \frac{5}{12}-\tan ^{-1} \frac{1}{239}$

$=\tan ^{-1} \frac{2 \cdot \frac{5}{12}}{1-\left(\frac{5}{12}\right)^{2}}-\tan ^{-1} \frac{1}{239} \quad\left(\because 2 \tan ^{-1} x=\tan ^{-1} \frac{2 x}{1-x^{2}}\right)$

$=\tan ^{-1} \frac{144 \times 5}{119 \times 6}-\tan ^{-1} \frac{1}{239}$

$=\tan ^{-1} \frac{120}{119}-\tan ^{-1} \frac{1}{239}$

$=\tan ^{-1} \frac{\frac{120}{119}-\frac{1}{239}}{1+\frac{120}{119} \cdot \frac{1}{239}} \quad\left(\because \tan ^{-1} x-\tan ^{-1} y=\tan ^{-1} \frac{x-y}{1+x y}\right)$

$=\tan ^{-1} \frac{120 \times 239-119}{119 \times 239+120}=\tan ^{-1} \frac{28680-119}{28441+120}$

$=\tan ^{-1} \frac{28561}{28561}=\tan ^{-1} 1=\frac{\pi}{4}$

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