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Trigonometric Functions — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsTrigonometric Functions2 Marks
MCQ
The value of $\sin \left[\cot ^{-1}\left(\tan \cos ^{-1} x\right)\right]$ is equal to
  • ✓
    $x$
  • B
    $\frac{\pi}{2}$
  • C
    $1$
  • D
    $\pi$
✓

Answer

Correct option: A.
$x$
(A) Let $\cos ^{-1} x=\theta \Rightarrow x=\cos \theta \Rightarrow \sec \theta=\frac{1}{x}$
$\Rightarrow \tan \theta=\sqrt{\sec ^2 \theta-1}=\sqrt{\frac{1}{x^2}-1}=\frac{1}{x} \sqrt{1-x^2}$
Now,
$\sin \left[\cot ^{-1}(\tan \theta)\right]=\sin \left[\cot ^{-1}\left(\frac{1}{x} \sqrt{1-x^2}\right)\right]$
Again, putting $x=\sin \theta$
$\therefore \quad \sin \cot ^{-1}\left(\frac{1}{x} \sqrt{1-x^2}\right)=\sin \cot ^{-1}\left(\frac{\sqrt{1-\sin ^2 \theta}}{\sin \theta}\right)$
$\begin{array}{l}=\sin \cot ^{-1}(\cot \theta) \\ =\sin \theta=x\end{array}$

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