Question

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

Answer 1

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

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

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

$=\frac{\sqrt{3}}{2}\times \frac{1}{3}-\frac{1}{2}\times \frac{2\sqrt{2}}{3}$

$=\frac{\sqrt{3}-2\sqrt{2}}{6}$