If cosθ + secθ = 2, then co$s^5$θ + se$c^5$θ equals:

Writing $c$ = cosθ, $c+1/c=2$ gives $c^2-2c+1=0$, so $(c-1{)}^2=0$ and cosθ = 1. Then co$s^5$θ + se$c^5$θ = 1 + 1 = 2. (Raising the given equation to the fifth power to get 32 is the trap.)