I'm doing Fourier series in class. I know that cos(nπ)=(-1)ⁿ and sin(nπ)=0, but today I came across the term cos(nπ/2) for n={0,1,2...}. What is this equivalent to?

cos (n*pi/2) is equal to 0 for all values of n. The link uses degrees as the angle measure, but 90 degrees is the same angle as pi/2 radians, and 270 degrees is the same as 3*pi/2 radians. Both have a cosine of 0. On a unit circle, the points referred to are on the Y-axis, and have an X-coordinate of 0. As the value of n in the expression goes up, the point travels around the circle over and over, always being on the X-axis, so the value of cosine is always 0.

http://www.easycalculation.com/trigonometry/trigonometry-tables.php

Not when n is even. When n is 2, for instance, it's cos(pi) which is -1, and when n is 4, it's cos(2*pi) which is 1. It's only zero when n is odd, and when n is even it alternates between 1 and -1. I'm looking for an expression (not involving sine or cosine) that sums this up.

Got it. It's (1/2)(1+(-1)?)(-1)^(n/2).

Plugging in n={0,1,2,3...} will give, {1,0,-1,0,1,0,-1...} which is what I needed.