Question

The value of $\cos ({\sin }^{-1}\frac{3}{5}+{\sin }^{-1}\frac{5}{13})$ is ____

• A. $-33/65$
• B. $+33/65$
• C. $-56/65$
• D. $+56/65$

Answer 1

Answer: (B)

Given:

$\cos ({\sin }^{-1}\frac{3}{5}+{\sin }^{-1}\frac{5}{13})$

We know that, ${\sin }^{-1}A+{\sin }^{-1}B={\sin }^{-1}x\sqrt{1-y^2}+y\sqrt{1-x^2}$

$=\cos \left({\sin }^{-1}(\frac{3}{5}\sqrt{1-{\left(\frac{5}{13}\right)}^2}+\frac{5}{13}\sqrt{1-{\left(\frac{3}{5}\right)}^2})\right)$
$=\cos \left({\sin }^{-1}(\frac{3}{5}\times \frac{12}{13}+\frac{5}{13}\times \frac{4}{5})\right)$
$=\cos \left({\sin }^{-1}\left(\frac{36+20}{65}\right)\right)=\cos \left({\sin }^{-1}\left(\frac{56}{65}\right)\right)$---(1)

We know that, ${\sin }^{-1}x={\cos }^{-1}\left(\sqrt{1-x^2}\right)$

Now, ${\sin }^{-1}\left(\frac{56}{65}\right)$ $={\cos }^{-1}\left(\sqrt{1-(\frac{56}{65}{)}^2}\right)={\cos }^{-1}\left(\frac{33}{65}\right)$

From (1)

$\cos \left({\sin }^{-1}\left(\frac{56}{65}\right)\right)=\cos \left({\cos }^{-1}\left(\frac{33}{65}\right)\right)=\frac{33}{65}$

Therefore, $\cos ({\sin }^{-1}\frac{3}{5}+{\sin }^{-1}\frac{5}{13})=\frac{33}{65}$

Hence, Option $C.$ is correct.