Calculating Long Binomial Expression

Question

Hello everyone how can I calculate this expression? $\binom {90}1(cos1)^{89}sin1-\binom {90}3(cos1)^{87}(sin1)^3+\binom {90}5(cos1)^{85}(sin1)^5...-\binom {90}{87}(cos1)^{3}(sin1)^{87}+\binom {90}{89}(cos1)(sin1)^{89}$

I tried to convert this expression to something like $(x+y)^n = \sum^n_{k=0}{\binom {n}k\cdot x^k\cdot y^{n-k}}$ but I didn't success.

Answer 1

This is the imaginary part of $$\sum_{k=0}^{90} {90 \choose k} \cos(1)^{90-k} (i \sin(1))^k$$ so...