Let theta_1=arcsin(5/13)θ1=arcsin(513)
Then,
sintheta_1=5/13sinθ1=513
And
costheta_1=sqrt(1-sin^2theta_1)cosθ1=√1−sin2θ1
=sqrt(1-25/169)=√1−25169
=sqrt(144/169)=√144169
=14/13=1413
Let theta_2=arccos(7/25)θ2=arccos(725)
costheta_2=7/25cosθ2=725
sintheta_2=sqrt(1-cos^2theta_2)sinθ2=√1−cos2θ2
=sqrt(1-49/625)=√1−49625
=sqrt(576/625)=√576625
=24/25=2425
Therefore,
cos(arcsin(5/13)-arccos(7/25))cos(arcsin(513)−arccos(725))
=cos(theta_1-theta_2)=cos(θ1−θ2)
=costheta_1costheta_2+sintheta_1sintheta_2=cosθ1cosθ2+sinθ1sinθ2
=14/13*7/25+5/13*24/25=1413⋅725+513⋅2425
=218/325=218325