Question

$\sin ({\cot }^{-1}x)=$

• A. $\sqrt{1+x^2}$
• B. $x$
• C. ${\left(1+x^2\right)}^{-\frac{3}{2}}$
• D. ${\left(1+x^2\right)}^{-\frac{1}{2}}$

Answer 1

The correct option is D

${\left(1+x^2\right)}^{-\frac{1}{2}}$

Explanation for the correct answer:

Solving the given expression by using inverse trigonometric identities:

Let us consider the following

$$\begin{gathered} {\cot }^{-1}x=\theta \\ \therefore x=cot\theta \end{gathered}$$

We know that,

$$\begin{gathered} \cos ec\theta =\sqrt{1–{\cot }^2\theta } \\ =\sqrt{1+x^2}[\because \mathrm{co}t\theta =x] \end{gathered}$$

We know that, $\sin \theta =\frac{1}{\cos ec\theta }$

$\therefore \sin \theta =\frac{1}{\sqrt{1+x^2}}$

$\Rightarrow \theta ={\sin }^{-1}\left(\frac{1}{\sqrt{1+x^2}}\right)$

Now, substitute the calculated values in the given function.

$$\begin{gathered} \Rightarrow \sin ({\cot }^{-1}x) \\ \Rightarrow \sin \left[{\sin }^{-1}\left(\frac{1}{\sqrt{1+x^2}}\right)\right] \\ \Rightarrow \frac{1}{\sqrt{1+x^2}}\left[\because \sin \left({\sin }^{-1}x\right)=x\right] \\ \Rightarrow {\left(1+x^2\right)}^{-\frac{1}{2}} \end{gathered}$$

Therefore, the correct answer is option (D).