The number of real angles θ for which si$n^4$θ + co$s^4$θ = 1/4 is:

si$n^4$θ + co$s^4$θ = 1 − 2si$n^2$θ·co$s^2$θ = 1 − (1/2)si$n^2$2θ. As si$n^2$2θ only ever runs from 0 to 1, the expression only ever runs from 1 down to 1/2. The target 1/4 lies below that whole range, so no angle satisfies the equation — the count is 0, not merely 'a few'. The value 1/4 is what si$n^4$θ + co$s^4$θ would be if one wrongly halved the minimum 1/2.