An acute angle θ satisfies sinθ·cosθ = 1/4. The value of si$n^6$θ + co$s^6$θ is:

$si$$n^6$θ + $co$$s^6$θ = $(si{n}^2\theta +co{s}^2\theta {)}^3-3si$$n^2$θ·$co$$s^2$θ$(si$$n^2$θ + $co$$s^2$θ) = 1 − 3$(sin\theta \cdot cos\theta {)}^2=1-3/16=13/16$. Here sinθ·cosθ = 1/4 means sin2θ = 1/2, $i$.e. θ = 15°, and $0.2588^6+0.9659^6=0.8125$ confirms it. Using the fourth-power formula $1-2p^2$ gives 7/8, dropping the 3 gives 15/16, and forgetting to square the product gives 1/4.