Question

The value of ${\sin }^{-1}x+{\cos }^{-1}x,\forall x\in [-1,1]$ is

• A. $\frac{\pi }{2}$
• B. $\frac{-5\pi }{3}$
• C. $\frac{-3\pi }{2}$
• D. $\pi$

Answer 1

The correct option is A $\frac{\pi }{2}$

Consider the right angle triangle $ABC$, $\angle C=\frac{\pi }{2}$
and hypotenuse $AC=1$ unit
$\angle A+\angle B+\angle C=\pi$
$\Rightarrow \theta +\frac{\pi }{2}+\varphi =\pi$
$\therefore \theta +\varphi =\frac{\pi }{2}$ .........eq(i)

$\sin \theta =xand\cos \varphi =x$
$\therefore \theta ={\sin }^{-1}xand\varphi ={\cos }^{-1}x$ ..........eq(ii)
using eq (i) and (ii)
${\sin }^{-1}x+{\cos }^{-1}x=\frac{\pi }{2}$