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  4. Number of values of x satisfying the equation cos ((4 π/3)- cos -1 x)=x, is

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

Inverse Trigonometric Functions

Solution:

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