Question

The value of ${\sin }^{-1}\left(\frac{12}{13}\right)-{\sin }^{-1}\left(\frac{3}{5}\right)$ is equal to :

• A. $\frac{\pi }{2}-{\cos }^{-1}\left(\frac{9}{65}\right)$
• B. $\pi -{\sin }^{-1}\left(\frac{63}{65}\right)$
• C. $\pi -{\cos }^{-1}\left(\frac{33}{65}\right)$
• D. $\frac{\pi }{2}-{\sin }^{-1}\left(\frac{56}{65}\right)$

Answer 1

The correct option is D $\frac{\pi }{2}-{\sin }^{-1}\left(\frac{56}{65}\right)$

${\sin }^{-1}\left(\frac{12}{13}\right)-{\sin }^{-1}\left(\frac{3}{5}\right)$
${\sin }^{-1}(\frac{12}{13}\sqrt{1-{\left(\frac{3}{5}\right)}^2}-\frac{3}{5}\sqrt{1-{\left(\frac{12}{13}\right)}^2})$
${\sin }^{-1}(\frac{12}{13}\times \frac{4}{5}-\frac{3}{5}\times \frac{5}{13})={\sin }^{-1}\left(\frac{33}{65}\right)={\cos }^{-1}\sqrt{1-{\left(\frac{33}{65}\right)}^2}={\cos }^{-1}\left(\frac{56}{65}\right)=\frac{\pi }{2}-{\sin }^{-1}\left(\frac{56}{65}\right)$