1977 AMC 12 Question 16

If $i^2=-1,$ then the sum $$
\begin{aligned} &\cos45^\circ+i\cos135^\circ+\cdots\\ &\quad+i^n\cos(45+90n)^\circ+\cdots\\ &\quad+i^{40}\cos3645^\circ \end{aligned}
$$ equals