Proving Taylor's expansion of sin x

Question

I am currently trying to prove Taylor's theorem of sin(x) using $(\cos x+i\sin x)^n$. I have managed to progress quite a bit, however, I am wondering how to proceed with the following: ${n \choose 2k+1}(\sin\theta)^{(2k+1)}=\frac{n(n-1)(n-2)...(n-2k)}{(2k+1)(2k)...1}(\sin\theta)^{(2k+1)}$ Also, from prev. calculations I have that n tends to infinity and n,k are both natural numbers where k starts from 0.Also, I assumed that n is odd earlier.Finally, I wish to prove it without prior knowledge of Taylor series of cos(x) and sin(x) ( i.e. I can't use ${e^{ix}=\cos(x)+isin(x)}$). Thank you for the help in advance!