Question

Find the value of ${\sin }^{-1}(2{\cos }^{2}x-1)+{\cos }^{-1}(1-2{\sin }^{2}x)$.

• A. $\frac{\pi }{2}$
• B. $\frac{\pi }{3}$
• C. $\frac{\pi }{4}$
• D. $\frac{\pi }{6}$

Answer 1

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

As we know that

$\cos 2A=2{\cos }^{2}A-1=1-2{\sin }^{2}A$

And also ${\text{sin}}^{-1}y+{\text{cos}}^{-1}y=\frac{\pi }{2}$

So ${\text{sin}}^{-1}(2{\cos }^{2}x-1)+{\text{cos}}^{-1}(1-2{\sin }^{2}x)={\text{sin}}^{-1}\left(\cos 2x\right)+{\text{cos}}^{-1}\left(\cos 2x\right)=\frac{\pi }{2}$