Explain two different ways to solve for the derivative of s(θ) = 200(cos^2)θ−θ100.

Answer 1

not sure what "θ1000" means, and usually one writes cos^2(θ)
In any case, aside from using first principles, once you have learned the rules for derivatives, I can only think of one way to find the desired derivative: just apply those rules.
d/dθ 200 cos^2(θ) = 200 * 2cosθ (-sinθ)
d/dθ θ100 = ???