Powers of Sine and Cosine

Let $f(x) = \sin^n x + \cos^n x$, where $n \in \mathbb{N}^*$.

1. When $n = 4$, determine whether $f(x)$ is even, odd, or neither. 2. For even $n$, the equation $f(x) = \dfrac{1}{2025}$ has a solution. Find the minimum such $n$. 3. If there exists $n$ such that the inequality

holds for all $x$, find the range of possible values of the real number $a$.