Question

The value of $\mathrm{x}$ which satisfies the equation ${\tan }^{-1}\mathrm{x}={\sin }^{-1}\left(\frac{3}{\sqrt{10}}\right)$ is

• A. $3$
• B. $-3$
• C. $\frac{1}{3}$
• D. $\frac{-1}{3}$

Answer 1

The correct option is A

$3$

Explanation for the correct option:

Given: ${\tan }^{-1}\mathrm{x}={\sin }^{-1}\left(\frac{3}{\sqrt{10}}\right)-\left(1\right)$

Let $\mathrm{A}={\sin }^{-1}\left(\frac{3}{\sqrt{10}}\right)$

$$\begin{gathered} \Rightarrow \sin A=\frac{3}{\sqrt{10}} \\ \Rightarrow \cos A=\sqrt{1-{\left(\frac{3}{\sqrt{10}}\right)}^2}\left[{\sin }^{2}A+{\cos }^{2}A=1\right] \\ \Rightarrow \cos A=\sqrt{1-\frac{9}{10}} \\ \Rightarrow \cos A=\sqrt{\frac{10-9}{10}} \\ \Rightarrow \cos A=\frac{1}{\sqrt{10}} \end{gathered}$$

$$\begin{gathered} \mathrm{tanA}=\frac{\mathrm{sinA}}{\mathrm{cosA}}=\frac{{\displaystyle \raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$\sqrt{10}$}\right.}}{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\sqrt{10}$}\right.}}=3 \\ A={\tan }^{-1}3 \\ \therefore {\sin }^{-1}\left(\frac{3}{\sqrt{10}}\right)={\tan }^{-1}3-\left(2\right) \end{gathered}$$

From equations $\left(1\right)$ and $\left(2\right)$

$$\begin{gathered} {\tan }^{-1}\left(x\right)={\tan }^{-1}\left(3\right) \\ x=3 \end{gathered}$$

Hence option A is the correct option.