7. The minimum value of the function $f(x)=\sin ^{2 k} x+\cos ^{2 k} x\left(k \in \mathbf{N}_{+}\right)$ is . $\qquad$
7. $\left(\frac{1}{2}\right)^{k-1}$.

Let $\sin ^{2} x=t$. Then $0 \leqslant t \leqslant 1$, and
$$
f(x)=g(t)=t^{k}+(1-t)^{k} \text {. }
$$

Notice that
$$
g^{\prime}(t)=k t^{k-1}-k(1-t)^{k-1}\left(k \in \mathbf{N}_{+}\right) \text {. }
$$

Then when $0<t<\frac{1}{2}$, $g^{\prime}(t)<0$; when $\frac{1}{2}<t<1$, $g^{\prime}(t)>0$.

Thus, $g(t)$ is a decreasing function on the interval $\left[0, \frac{1}{2}\right]$ and an increasing function on $\left[\frac{1}{2}, 1\right]$.
Therefore, the minimum value of $f(x)$ is $g\left(\frac{1}{2}\right)=\left(\frac{1}{2}\right)^{k-1}$.