We know that the angles are greater than 0° and less than 90°.  So
their sum must be greater than 0° and less than 270°.
We therefore know that the equation above will be true if and only
if the sine of the left side is 0, because 180° is the only angle
between 0° and 270° with a sine of 0.
  
Draw the three right triangles below. The angles marked in red are
the three angles, A=arcsin(12/13),  B=arccos(4/5), and C=arctan(63/16).
I got the missing sides by using the Pythagorean theorem.  I
assume you understand that part.  If not, tell me in the thank you
note form below, and I'll get back to you by email.



We use the double angle formulas: 

sin(X+Y)=sinXcosY+cosXsinY

and 

cos(X+Y)=cosXcosY-sinXsinY

We start with the sine of the left side and hope to get 0:

sin(A+B+C) = sin[(A+B)+C] = sin(A+B)cosC+cos(A+B)sinC =

(sinAcosB+cosAsinB)cosC+(cosAcosB-sinAsinB)sinC =

Now we use the adjacent, opposite, and hypotenuse in the
above 3 right triangles to substitute:






 




So we've proved it.

Edwin