Question

Consider $f\left(x\right)={\sin }^{–1}\left(\sin x\right)-{\cos }^{–1}\left(\cos x\right).$ Then which of the following statements is/are TRUE?

• A. $f\left(x\right)=0$ for $x\in (0,\frac{\pi }{4})$
• B. $f\left(x\right)<0$ for $x\in (\frac{\pi }{2},\frac{3\pi }{4})$
• C. $f\left(x\right)>0$ for $x\in (\pi ,\frac{5\pi }{4})$
• D. $f\left(x\right)<0$ for $x\in (-\pi ,-\frac{3\pi }{4})$

Answer 1

Answer: (A), (B), (D)

$f\left(x\right)<0$ for $x\in (-\pi ,-\frac{3\pi }{4})$

We know, the graph of ${\sin }^{-1}\left(\sin x\right)$


And the graph of ${\cos }^{-1}\left(\cos x\right)$


For $x\in (0,\frac{\pi }{2}),$
$f\left(x\right)=x-x=0$

For $x\in (\frac{\pi }{2},\pi ),$
$f\left(x\right)=\pi -x-x=\pi -2x<0$

For $x\in (\pi ,\frac{3\pi }{2}),$
$f\left(x\right)=\pi -x-(2\pi -x)=-\pi <0$

For $x\in (-\pi ,\frac{-\pi }{2}),$
$f\left(x\right)=-(\pi +x)-(-x)=-\pi <0$