Euler's Formula
#e^{i theta}=cos theta + i sin theta#
Let us first review some useful power series.
#e^x=1/{0!}+x/{1!}+x^2/{2!}+cdots#
#cos x=1/{0!}-x^2/{2!}+x^4/{4!}-cdots#
#sin x=x/{1!}-x^3/{3!}+x^5/{5!}-cdots#
Now, we are ready to prove Euler's Formula.
Proof
By rewriting as a power series,
#e^{i theta}=1/{0!}+(i theta)/{1!}+(itheta)^2/{2!}+(i theta)^3/{3!}+(i theta)^4/{4!}+(i theta)^5/{5!}+cdots#
by distributing the powers,
#=1/{0!}+i theta/{1!}+i^2 theta^2/{2!}+i^3 theta^3/{3!}+i^4 theta^4/{4!}+i^5 theta^5/{5!}+cdots#
by #i^2=-1#
#=1/{0!}+i theta/{1!}-theta^2/{2!}-i theta^3/{3!}+theta^4/{4!}+i theta^5/{5!}-cdots#
by separating the real part and the imaginary part,
#=(1/{0!}-theta^2/{2!}+theta^4/{4!}-cdots)+i(theta/{1!}-theta^3/{3!}+theta^5/{5!}-cdots)#
by identifying the power series,
#=cos theta + i sin theta#
Hence, we have Euler's Formula
#e^{i theta}=cos theta+i sin theta#.
I hope that this was helpful.
