Question

The minimum value of $x$ which satisfies the inequality $\left(s i n^{- 1} x\right)^{2}\geq \left(c o s^{- 1} x\right)^{2}$ is

• A. $\frac{1}{\sqrt{2}}$
• B. $\frac{1}{2}$
• C. $\frac{\sqrt{3}}{2}$
• D. $\frac{1}{\sqrt{3}}$

Answer 1

Answer: (A)

$\left(s i n^{- 1} x\right)^{2}-\left(c o s^{- 1} x\right)^{2}\geq 0$
$\left(s i n^{- 1} x + c o s^{- 1} x\right)\left(s i n^{- 1} x - c o s^{- 1} x\right)\geq 0$
$\Rightarrow sin^{- 1}x\geq cos^{- 1}x$
Clearly from graph
$x\in \left[\frac{1}{\sqrt{2}} , 1\right]$