Question

Evaluate$:\sin \left(2{\cos }^{-1}(-\frac{3}{5})\right)$

Answer 1

$\sin \left(2{\cos }^{-1}\left(\frac{3}{5}\right)\right){\cos }^{-1}\frac{3}{5}={\sin }^{-1}\frac{4}{5}2{\sin }^{-1}x={\sin }^{-1}x\sqrt{1-x^2}2{\sin }^{-1}\left(\frac{4}{5}\right)={\sin }^{-1}[\frac{-4}{5}\times \frac{3}{5}]={\sin }^{-1}\left(\frac{-12}{25}\right)\sin \left(2{\cos }^{-1}\left(\frac{-3}{5}\right)\right)=\sin \left({\sin }^{-1}\left(\frac{-12}{25}\right)\right)=\frac{-12}{25}$