Mathematics - Trigonometric Ratios & Identities Question with Solution | TestHub

Let $g(x)={\sin }^{-1}|\sin x|+{\cos }^{-1}(\cos x)=\{\begin{array}{ccc}2x& ;& 0\le x\le \frac{\pi }{2}\\ \pi & ;& \frac{\pi }{2}<x\le \frac{3\pi }{2}\\ 4\pi -2x& ;& \frac{3\pi }{2}<x\le 2\pi \end{array}$.

$g(x)$ is periodic with period $2\pi$ and is constant in the interval $\left[2n\pi +\frac{\pi }{2},2n\pi +\frac{3\pi }{2}\right]$, $n\in I$.

Now, $f(x)=g(\alpha x)$.

$$\begin{array}{rl}& \therefore f(x)\text{ is constant in }\left[\frac{2n\pi }{\alpha }+\frac{\pi }{2\alpha },\frac{2n\pi }{\alpha }+\frac{3\pi }{2\alpha }\right]\\ & \therefore 2\pi =\frac{3\pi }{2\alpha }-\frac{\pi }{2\alpha }\\ & \Rightarrow \alpha =\frac{1}{2}\end{array}$$