The minimum value of se$c^2$θ + cose$c^2$θ is:

se$c^2$θ + cose$c^2$θ = $(si$$n^2$θ + $co$$s^2$θ)/$(si$$n^2$θ·$co$$s^2$θ) $=1/(sin\theta \cdot cos\theta {)}^2$. Since |sinθ·cosθ| is at most 1/2, the expression is at least 1/(1/4) = 4, attained at θ = 45°.