d/dx e^ix = ie^ix
-----------------
d/dx cosx + isinx
= -sinx + icosx
= i (cosx + isinx)
-----------------
Both functions satisfy the differential equation dy/dx = iy
with initial value (x=0)equal to one
Thus they must be equal ( if two functions are the same at one point and change in the same way, their entire functions must be equal.)
Thus we have again, e^ix = cosx + isinx
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