Question

The value of $\cos [{\sin }^{-1}\left(\frac{-4}{5}\right)-{\cos }^{-1}\left(\frac{4}{5}\right)]=$

• A. $\frac{\pi }{2}$
• B. $-\frac{\pi }{2}$
• C. $0$
• D. $\pi$

Answer 1

The correct option is C $0$

Given : $\cos [{\sin }^{-1}\left(\frac{-4}{5}\right)-{\cos }^{-1}\left(\frac{4}{5}\right)]$
$=\cos [-{\sin }^{-1}\left(\frac{4}{5}\right)-{\cos }^{-1}\left(\frac{4}{5}\right)]$
$=\cos [-({\sin }^{-1}\left(\frac{4}{5}\right)+{\cos }^{-1}\left(\frac{4}{5}\right))]$
$=\cos [{\sin }^{-1}\left(\frac{4}{5}\right)+{\cos }^{-1}\left(\frac{4}{5}\right)]$
$=\cos \left(\frac{\pi }{2}\right)=0$ $(\because {\sin }^{-1}(x)+{\cos }^{-1}(x)=\frac{\pi }{2})$ for $x\in [-1,1]$