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Answer 1 · 2016-05-16T23:45:32

$sin(tan^-1 (3/4)+cos^-1 (5/13)) = (3/5)(5/13)+(4/5)(12/13) = 63/65$

$tan^-1(3/4)$ is a $theta$ between $-pi/2$ and $pi/2$ with $tan theta = 3/4$

$cos^-1(5/13)$ is a $phi$ between $0$ and $pi$ with $cos phi = 5/13$

We have been asked to find $sin(theta + phi)$ .

We know that $sin(theta + phi) = sin theta cos phi + cos theta sin phi$

We know that $tan theta = 3/4$ and $0 < theta < pi/2$ , we can find that $sin theta = 3/5$ and $cos theta = 4/5$

We know that $cos phi = 5/13$ and $0 < phi < pi/2$ so we find $sin phi = 12/13$ .

Therefore

$sin(tan^-1 (3/4)+cos^-1 (5/13)) = (3/5)(5/13)+(4/5)(12/13) = 63/65$ .