For 0° < θ < 90°, the expression si$n^8$θ + co$s^8$θ takes its smallest value when θ equals:

Let $u$ = si$n^2$θ·co$s^2$θ, which runs over (0, 1/4] on this interval. Then si$n^8$θ + co$s^8$θ = $(1-2u{)}^2-2u^2=1-2u(2-u)$, and since $2-u$ is positive the whole expression falls steadily as $u$ rises. So the least value comes from the largest $u$, namely $u=1/4$ at sinθ = cosθ, i.e. θ = 45°, where the value is $2\times (1/\surd 2{)}^8=1/8$. At 30° and 60° the value is the same, 0.0039 + 0.3164 = 0.3203 — far above 1/8 = 0.125 — and 0° is outside the stated range.