Question

If $f(x)=\left(\frac{\pi }{2}-{\cos }^{-1}(\cos x)\left(\frac{\pi }{2}-{\sin }^{-1}(\sin x)\right)\right)$, where $x\in [-2\pi ,2\pi ]$, then the number of integers in the range of $f(x)$ is

Answer 1

Answer: 5

Periodic function with period $=2\pi$
$f\left(x\right)={\left(sin\right)}^{-1}\left(cosx\right)\cdot {\left(cos\right)}^{-1}\left(sinx\right)$
$f\left(x\right)=\{\begin{array}{ll}{\left(\frac{\pi }{2}-x\right)}^2& ,x\in \left[0,\frac{\pi }{2}\right]\\ -{\left(\frac{\pi }{2}-x\right)}^2& ,x\in \left(\frac{\pi }{2},\pi \right)\\ \left(x-\frac{\pi }{2}\right)\left(x-\frac{3\pi }{2}\right)& ,x\in \left[\pi ,\frac{3\pi }{2}\right]\\ \left(x-\frac{3\pi }{2}\right)\left(\frac{5\pi }{2}-x\right)& ,x\in \left[\frac{3\pi }{2},2\pi \right]\end{array}$

$R_f\in \left[-\frac{{\pi }^2}{4},\frac{{\pi }^2}{4}\right]$