Question

$\cos ({\cos }^{-1}\frac{1}{3}+{\cos }^{-1}\frac{1}{5})+\cos ({\sin }^{-1}\frac{1}{3}+{\sin }^{-1}\frac{1}{5})=$__________

• A. $0$
• B. $\pi$
• C. $\frac{\pi }{2}$
• D. $\frac{\pi }{4}$

Answer 1

The correct option is A $0$

${\cos }^{-1}x+{\sin }^{-1}x=\frac{\pi }{2}$

$⟹{\cos }^{-1}x=\frac{\pi }{2}-{\sin }^{-1}x$

and ${\sin }^{-1}x=\frac{\pi }{2}-{\cos }^{-1}x$

Now-

${\cos }^{-1}\frac{1}{3}+{\cos }^{-1}\frac{1}{5}$

$=\frac{\pi }{2}-{\sin }^{-1}\frac{1}{3}+\frac{\pi }{2}-{\sin }^{-1}\frac{1}{5}$

$=\pi -{\sin }^{-1}\frac{1}{3}-{\sin }^{-1}\frac{1}{5}$

$⟹\cos ({\cos }^{-1}\frac{1}{3}+{\cos }^{-1}\frac{1}{5})=\cos \{\pi -({\sin }^{-1}\frac{1}{3}+{\sin }^{-1}\frac{1}{5})\}$

$⟹\cos ({\cos }^{-1}\frac{1}{3}+{\cos }^{-1}\frac{1}{5})=-\cos ({\sin }^{-1}\frac{1}{3}+{\sin }^{-1}\frac{1}{5})$

$⟹\cos ({\cos }^{-1}\frac{1}{3}+{\cos }^{-1}\frac{1}{5})+\cos ({\sin }^{-1}\frac{1}{3}+{\sin }^{-1}\frac{1}{5})=0$