Question

If $({\cos }^{-1}x{)}^2-({\sin }^{-1}x{)}^2>0$, then

• A. $x<\frac{1}{2}$
• B. $-1<x\sqrt{2}$
• C. $0\le x<\frac{1}{\sqrt{2}}$
• D. $-1\le x<\frac{1}{\sqrt{2}}$

Answer 1

Answer: (D)

${\left(co{s}^{-1}x\right)}^2-{\left(si{n}^{-1}x\right)}^2>0$
$\Rightarrow \left(co{s}^{-1}x+si{n}^{-1}x\right)\left(co{s}^{-1}x-si{n}^{-1}x\right)>0$
$\Rightarrow \frac{\pi }{2}\left(\frac{\pi }{2}-2si{n}^{-1}x\right)>0$
$\Rightarrow \frac{\pi }{2}>2si{n}^{-1}x$
$\Rightarrow \frac{\pi }{4}>si{n}^{-1}x$ i.e., $si{n}^{-1}x<\frac{\pi }{4}$
Also $-\frac{\pi }{2}\le si{n}^{-1}x\le \frac{\pi }{2}$
$\therefore -\frac{\pi }{2}\le si{n}^{-1}x<\frac{\pi }{4}$
$\Rightarrow -1\le x<\frac{1}{\sqrt{2}}$