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Trigonometric Functions — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsTrigonometric Functions2 Marks
MCQ
$\sin \left(\cot ^{-1} x\right)=$
  • A
    $\sqrt{1+x^2}$
  • B
    $x$
  • C
    $\left(1+x^2\right)^{\frac{-1}{2}}$
  • ✓
    $\left(1+x^2\right)^{\frac{-1}{2}}$
✓

Answer

Correct option: D.
$\left(1+x^2\right)^{\frac{-1}{2}}$
(D) Let $\cot ^{-1} x=\theta \Rightarrow x=\cot \theta$
Now $\operatorname{cosec} \theta=\sqrt{1+\cot ^2 \theta}=\sqrt{1+x^2}$
$\therefore \quad \sin \theta=\frac{1}{\operatorname{cosec} \theta}=\frac{1}{\sqrt{1+x^2}}$
$\Rightarrow \theta=\sin ^{-1} \frac{1}{\sqrt{1+x^2}}$
$\therefore \quad \sin \left(\cot ^{-1} x\right)=\sin \left(\sin ^{-1} \frac{1}{\sqrt{1+x^2}}\right)$
$=\frac{1}{\sqrt{1+x^2}}$
$=\left(1+x^2\right)^{\frac{-1}{2}}$

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