Which of these numbers can be the value of sinθ + cosecθ for some real angle θ?

Write $x$ = sinθ, so the expression is $x+1/x$ with 0 < |$x$| ≤ 1. For positive $x$ the sum is at least 2 (equality only at $x=1$), and for negative $x$ it is at most −2, so no value strictly between −2 and 2 is ever reached. Only 5/2 clears the gap: $x=1/2$ gives 1/2 + 2 = 5/2, i.e. θ = 30°. The other three all sit inside the forbidden band, and 0 would additionally require sinθ and cosecθ to have opposite signs, which is impossible.