Explanation:

Using algebraic maxima finding technique we transform part of the given expression to a square of sums.
sin⁴θ+cos⁴θ
=(sin²θ+cos²θ)2−2sin²θcos²θ,
=1−2sin²θcos²θ.
The maximum of this expression can only be 1 when, the second term is zero, or when either sinθ or cosθ is 0.
This is a quick method as it has used the trigonometric relations along with the algebraic maxima technique elegantly.