Using the fact that sin(A+B)=sin A cos B + cos A sin B and differentiation,
obtain the sum formula for cosines.
Using sin (A+B) = sin A cos B + cos A sin B, we have on differentiating w.r.t. x, regarding B as constant and consider A as a formula of x, we have
cos(A+B)dAdx=cos B.cos AdAdx−(sin A dAdx)sin B⇒ cos(A+B)dAdx=(cos A cosB−sin A sin B)dAdx⇒ cos(A+B)=cos A cos B−sin A sin B (∵ dAdx≠0)