Question

If ${\sin }^{-1}\frac{3}{5}+{\cos }^{-1}\frac{12}{13}={\sin }^{-1}C,$ then $C=$

• A. $\frac{8}{65}$
• B. $\frac{24}{65}$
• C. $\frac{16}{65}$
• D. $\frac{56}{65}$

Answer 1

The correct option is D $\frac{56}{65}$

Given,
${\sin }^{-1}\frac{3}{5}+{\cos }^{-1}\frac{12}{13}={\sin }^{-1}C$
$\Rightarrow {\sin }^{-1}\frac{3}{5}+{\sin }^{-1}\frac{5}{13}={\sin }^{-1}C$
Using the formula:
$\{{\sin }^{-1}x+{\sin }^{-1}y={\text{sin}}^{-1}\{x\sqrt{1-y^2}+y\sqrt{1-x^2}\}\}$
$\Rightarrow {\sin }^{-1}\{\frac{3}{5}\sqrt{1-\frac{25}{169}}+\frac{5}{13}\sqrt{1-\frac{9}{25}}\}={\sin }^{-1}C$
$\Rightarrow C=\frac{36}{65}+\frac{4}{13}=\frac{56}{65}$