Evaluate cos(sin^(-1)(3/5) + sin^(-1)(5/13) + sin^(-1)(33/65)) (1) 0 (2) 1

Question

Evaluate

\(\cos \left(\sin ^{-1}\left(\frac{3}{5}\right)+\sin ^{-1}\left(\frac{5}{13}\right)+\sin ^{-1}\left(\frac{33}{65}\right)\right)\)

(1) 0

(2) 1

(3) \(\cos \frac{5}{13}\)

(4) 2

Answer 1

Correct option is (1) 0 

\(\cos \left(\sin ^{-1} \frac{3}{5}+\sin ^{-1} \frac{5}{13}+\sin ^{-1} \frac{33}{65}\right)\)

\(=\cos \left(\sin ^{-1} \frac{3}{5}+\sin ^{-1}\left[\frac{5}{13} \sqrt{\frac{1-33^2}{65^2}}+\frac{33}{65} \sqrt{1-\frac{5^2}{13^2}}\right]\right)\)  

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

\(=\cos \left(\sin ^{-1} \frac{3}{5}+\cos ^{-1} \frac{3}{5}\right)\)  

\(=\cos \left(\frac{\pi}{2}\right)=0\)