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Inverse Trigonometric Functions — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsInverse Trigonometric Functions1 Mark
MCQ
If $\tan^{-1}3+\tan^{-1}\text{x}=\tan^{-1}8,$ then $x =$
  • A
    $5$
  • ✓
    $\frac{1}{5}$
  • C
    $\frac{5}{14}$
  • D
    $\frac{14}{5}$
✓

Answer

Correct option: B.
$\frac{1}{5}$
We know that $\tan^{-1}\text{x}+\tan^{-1}\text{y}=\tan^{-1}\frac{\text{x}+\text{y}}{1-\text{xy}}$
Now, $\tan^{-1}3+\tan^{-1}\text{x}=\tan^{-1}8$
$\Rightarrow\tan^{-1}\Big(\frac{3+\text{x}}{1-3\text{x}}\Big)=\tan^{-1}8$
$\Rightarrow\frac{3+\text{x}}{1-3\text{x}}=8$
$\Rightarrow3+\text{x}=8-24\text{x}$
$\Rightarrow3-8=-24\text{x}-\text{x}$
$\Rightarrow-5=-25\text{x}$
$\Rightarrow\text{x}=\frac{5}{25}=\frac{1}{5}$

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