you can do it geometrically .By using two right triangles one of angle A and the other of angle B side by side and computing the sin(A +B ) od new TRIANGLE in terms of sine and cosines of A an B angles
You can also find it with a VECTOR approach and a ROTATION and find the MATRIX of such rotation it comes from the fact that ROTATIONS are linear mappings .so they verify :
R(A+ B) = R(A) + R(B) ..This is linear algebra is very simple to demostrate