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)
eulers derivation
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!+..

Now,
E(ix)=1+ix+(ix)^2/2!+(ix)^3/3!+(ix)^4/4!+(ix)^5/5!+…
E(ix)=(1-x^2/2!+x^4/4!-x^6/6!+…)+i(x-x^3/3!+x^5/5!-x^7/7!+…)
=cos(x)-i*sin(x)

the video for derivation is here

for derivation of Taylor’s series:

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