Question

Number of values of $x$ satisfying the equation $\cos \left(\frac{4\pi }{3}-{\cos }^{-1}x\right)=x$, is

Answer 1

Answer: 1

$x=\cos \left(\frac{4\pi }{3}-{\cos }^{-1}x\right)=\frac{-1}{2}x+\left(-\frac{\sqrt{3}}{2}\right)\sqrt{1-x^2}$
$\Rightarrow \frac{3}{2}x=-\frac{\sqrt{3-3x^2}}{2}.$
Squaring both sides yields
$9x^2=3-3x^2\Rightarrow 12x^2=3$
$\Rightarrow x=\pm \frac{1}{2}.(x=\frac{1}{2}$ rejected $)$
So, $x=\frac{-1}{2}$ is the only solution.