Evaluate:

Question:

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

Solution:

$\cos \left(\sin ^{-1} \frac{3}{5}+\sin ^{-1} \frac{5}{13}\right)=\cos \left\{\sin ^{-1}\left(\frac{3}{5} \sqrt{1-\left(\frac{5}{13}\right)^{2}}+\frac{5}{13} \sqrt{1-\left(\frac{3}{5}\right)^{2}}\right)\right\}$                                $\left[\because \sin ^{-1} x+\sin ^{-1} y=\sin ^{-1}\left(x \sqrt{1-y^{2}}+y \sqrt{1-x^{2}}\right)\right]$

$=\cos \left\{\sin ^{-1}\left(\frac{3}{5} \times \frac{12}{13}+\frac{5}{13} \times \frac{4}{5}\right)\right\}$

$=\cos \left\{\sin ^{-1}\left(\frac{36}{65}+\frac{4}{13}\right)\right\}$

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

$=\cos \left\{\cos ^{-1} \sqrt{1-\left(\frac{56}{65}\right)^{2}}\right\} \quad\left[\because \sin ^{-1} x=\cos ^{-1} \sqrt{1-x^{2}}\right]$

$=\cos \left\{\cos ^{-1} \frac{33}{65}\right\}$

$=\frac{33}{65}$

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