Question

Evaluate

$\cos \left({\sin }^{-1}\frac{1}{2}+{\cos }^{-1}\frac{1}{3}\right)$

• A. $\frac{\sqrt{3}-\sqrt{8}}{6}$
• B. $\frac{-\sqrt{3}+\sqrt{8}}{6}$
• C. $\frac{\sqrt{3}+\sqrt{8}}{6}$
• D. 0

Answer 1

The correct option is A $\frac{\sqrt{3}-\sqrt{8}}{6}$

Let $x={\text{sin}}^{-1}\frac{1}{2}⟹\sin x=\frac{1}{2}⟹\cos x=\frac{\sqrt{3}}{2}$

$y={\text{cos}}^{-1}\frac{1}{3}⟹\cos y=\frac{1}{3}⟹\sin x=\frac{2\sqrt{2}}{3}$

$\cos ({\text{sin}}^{-1}\frac{1}{2}+{\text{cos}}^{-1}\frac{1}{3})=\cos (x+y)=\cos x\cos y-\sin x\sin y=\frac{\sqrt{3}}{2}\times \frac{1}{3}-\frac{\sqrt{8}}{3}\times \frac{1}{2}=\frac{\sqrt{3}-\sqrt{8}}{6}$