Question

Let $x \in \left(0,1\right).$ The set of all x such that $sin^{-1} x >\, cos^{-1} x, $ is the interval:

• A. $\left(\frac{1}{2}, \frac{1}{\sqrt{2}}\right)$
• B. $\left(\frac{1}{\sqrt{2}}, 1\right)$
• C. (0,1)
• D. $\left(0,\frac{\sqrt{3}}{2}\right)$

Answer 1

Answer: (B)

Given $\,sin^{-1} x>\, cos^{-1} x$ where $\, x \in \left(0,1\right)$
$\Rightarrow sin^{-1} x>\, \frac{\pi}{2}-sin^{-1} x$
$\Rightarrow 2 sin^{-1} x>\, \frac{\pi}{2} \Rightarrow sin^{-1} x>\, \frac{\pi}{4}$
$\Rightarrow x>\, sin \frac{\pi}{4} \Rightarrow x>\, \frac{1}{\sqrt{2}}$
Maximum value of $sin^{-1} x \, is \, \frac{\pi}{2}$
So, maximum value of x is 1. So,
$x \in \left(\frac{1}{\sqrt{2}} , 1\right).$