Derivation of Euler’s forumula for imaginary numbers [e(x)=coz(x)+i*sin(x)]

Derivation of Euler’s forumula for imaginary numbers
a+ib=r(cosθ+ i sinθ)
From Tylor’s series:
F(x)=F(0)+x*f’(x)/1!+x^2*f’’(0)/2!+……+x^n*fn(0)/n!+….
Therefore,
Sin(x)=x-x^3/3!+x^5/5!-x^7/7!+….
Cos(x)=1-x^2/2!+x^4/4!-x^6/6!+….
e(x)=1+x+x^2/2!+x^3/3!+..