Question

The equation $2co{s}^{-1}x=si{n}^{-1}\left(2x\sqrt{1-x^2}\right)$ is valid for all values of x satisfying

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

Answer 1

The correct option is D $1/\sqrt{2}\le x\le 1$

If we denote $co{s}^{-1}x$ by $y$, then
since $0\le co{s}^{-1}x\le \pi \Rightarrow 0\le 2y\le 2\pi$ (1)
Also since $-\pi /2\le si{n}^{-1}\left(2x\sqrt{1-x^2}\right)\le \pi /2\Rightarrow -\pi /2\le si{n}^{-1}sin\left(2y\right)\le \pi /2$
$\Rightarrow -\pi /2\le 2y\le \pi /2$ (2)
From (1) and (2) we find $0\le 2y\le \pi /2$
$\Rightarrow 0\le y\le \pi /4\Rightarrow 0\le co{s}^{-1}x\le \pi /4$
which holds if $1/\sqrt{2}\le x\le 1$