Question

$\sin \left(2{\sin }^{-1}\sqrt{\frac{63}{65}}\right)=$

• A. $\frac{2\sqrt{126}}{65}$
• B. $\frac{4\sqrt{65}}{65}$
• C. $\frac{8\sqrt{63}}{65}$
• D. $\frac{\sqrt{63}}{65}$

Answer 1

The correct option is A $\frac{2\sqrt{126}}{65}$

$\sin \left(2{\sin }^{-1}\left(\sqrt{\frac{63}{65}}\right)\right)$

$=\sin ({\sin }^{-1}\sqrt{\frac{63}{65}}+{\sin }^{-1}\sqrt{\frac{63}{65}})$

$=\sin \left({\sin }^{-1}(\sqrt{\frac{63}{65}}\sqrt{\frac{2}{65}}+\sqrt{\frac{63}{65}}\sqrt{\frac{2}{65}})\right)$

$=\sin \left({\sin }^{-1}\left(2\frac{\sqrt{126}}{65}\right)\right)$ .... $[{\sin }^{-1}x+{\sin }^{-1}y={\sin }^{-1}[x\sqrt{1-y^2}+y\sqrt{1-x^2}\left]\right]$

$=\frac{2\sqrt{126}}{65}$

Hence, option A is correct.