As θ runs over all real values, the greatest value of sinθ + cos2θ is:

Write cos2θ = 1 − 2si$n^2$θ and put $s$ = sinθ: the expression becomes $1+s-2s^2=9/8-2(s-1/4{)}^2$, which peaks at $s=1/4$ — a perfectly legal sine — with value 9/8 = 1.125. The trap 2 assumes sinθ and cos2θ can both reach 1 at once, but cos2θ = 1 forces sinθ = 0, giving only 1.