Hi

It is very simple.

Lets first start with taylors series.

From taylors series for e^ix we have

e^ix = 1 + ix + (ix)^2/2! + (ix)^3/3! + (ix)^4/41 + ...

now put the value of i in the above equation it will get reduced to 

e^ix = cosx + isinx

It gets reduced to that because cosx and sinx reduction from taylors series is present in e^ix formulae

now we have 

e^ix = cosx + isinx

in the above equation put x = pie ( 3.147 ) and see the magic at work

cos(pie) = -1

sin(pie) = 0 

e^i(pie) + 1 = 0

the above equation is called as eulers equation.