Question

If $x+\frac{1}{x}=2$, then the principal value of ${\sin }^{-1}x$ is

• A. $\frac{\mathrm{\pi }}{4}$
• B. $\frac{\mathrm{\pi }}{2}$
• C. $\mathrm{\pi }$
• D. $\frac{3\mathrm{\pi }}{2}$

Answer 1

The correct option is B

$\frac{\mathrm{\pi }}{2}$

Explanation for the correct option.

Step 1. Solve the equation for $x$.

The given equation $x+\frac{1}{x}=2$ can be solved for $x$ as:

$$\begin{gathered} x+\frac{1}{x}=2 \\ \Rightarrow x^2+1=2x \\ \Rightarrow x^2-2x+1=0 \\ \Rightarrow {\left(x-1\right)}^2=0 \\ \Rightarrow x-1=0 \\ \Rightarrow x=1 \end{gathered}$$

Step 2. Find the value of ${\sin }^{-1}x$.

As the value of $x$ for the given equation is $1$. So in the expression ${\sin }^{-1}x$ substitute $1$ for $x$ and find its value.

$\begin{array}{rcl}{\sin }^{-1}x& =& {\sin }^{-1}1\\ & =& \frac{\mathrm{\pi }}{2}\left[\sin \frac{\mathrm{\pi }}{2}=1\right]\end{array}$

So for the equation $x+\frac{1}{x}=2$, the principal value of ${\sin }^{-1}x$ is $\frac{\mathrm{\pi }}{2}$.

Hence, the correct option is B.