The Euler formula can be used to cut out a huge amount of crap that is normally necessary to prove the summation of angle trigonometric identities (cos(A+B) etc) as well as the trigonometric derivatives.
Let's first state Euler.
cos A + i sin A = eiA
Now prove the summation formulae.
cos (A+B) + i sin (A+B) = ei(A+B)
= (cos A + i sin A)(cos B + i sin B)
= cos A cos B - sin A sin B + i (sin A cos B + cos A sin B)
Compare coefficients. Taking the real part of both sides we get:
cos (A+B) = cos A cos B - sin A sin B
Taking the imaginary part, we get:
sin (A+B) = sin A cos B + cos A sin B
Now prove the derivatives. This takes even fewer lines.
d/dA (cos A + i sin A) = d/dA eiA
= i cos A - sin A
Taking the real part
d/dA cos A = -sin A
d/dA sin A = cos A
Now why can't you use Euler to prove the derivative? Well because Euler comes from the Taylor expansions of sin, cos and exp, which uses the derivatives so we need to know the trigonometric derivatives before we can prove them in this way. Hence we'd be going round in circles.
See! I remember all this stuff! Who says I didn't?