For every angle θ, the value of si$n^2$θ + si$n^2$(90° − θ) + si$n^2$(180° + θ) + si$n^2$(270° − θ) is:

Reduce each allied angle: sin(90° − θ) = cosθ, sin(180° + θ) = −sinθ and sin(270° − θ) = −cosθ. Squaring wipes out both minus signs, so the four terms are si$n^2$θ, co$s^2$θ, si$n^2$θ and co$s^2$θ, which pair up into 1 + 1 = 2. Check at θ = 37°: 0.3624 + 0.6376 + 0.3624 + 0.6376 = 2.000. Counting each squared term as a full 1 gives 4; pairing only the first two and stopping gives 1; and 0 expects the minus signs to cancel the terms — but they are inside squares, so they never survive.