1. axiom i*i=0
---------------
   (a+ib)
  e      = a+ib , b->0

  result:

   (a+ib)
  e      = 1+ib ; b->0, i*i=0


2. axiom i*i=1
--------------
   (a+ib)
  e      = a+ib , b->0

  result:

   (a+ib)
  e      = cosh(x) +  i*sinh(x)+ib ; b->0, i*i=1

  sinh(x) + cosh(x) = e^x


3. axiom i*i=-1
--------------
   (a+ib)
  e      = a+ib , b->0

  result:

   (a+ib)
  e      = cos(x) +  i*sin(x)+ib ; b->0, i*i=-1



=>
                    _
sin(x) + cos(x) = \/2 * sin(x+pi/4)