Six. (20 points) In the sequence $\left\{a_{n}\right\}$,
$$
a_{n}=\cos 2^{n} \theta(n \in \mathbf{N}) \text {. }
$$
(1) If the set $A=\left\{\theta \mid a_{0}>0, a_{1}<0, \theta \in\right.$ $\mathbf{R}\}, B=[0,2 \pi]$, find $A \cap B$;
(2) Find all real numbers $\theta$, such that every term of the sequence $\left\{a_{n}\right\}$ is negative.
Let $a_{0}>0$, i.e., $\cos \theta>0$.
Solving gives $2 k \pi-\frac{\pi}{2}<\theta<2 k \pi+\frac{\pi}{2}(k \in \mathbf{Z})$.
Since $a_{n+1}=2 a_{n}^{2}-1<0$, we have $-1 \leqslant 2 a_{n}^{2}-1<0$, which implies $0 \leqslant a_{n}^{2}<\frac{1}{2}$.
Thus, $-\frac{\sqrt{2}}{2}<a_{n}<\frac{\sqrt{2}}{2}$.
Since $a_{n}<0$, we have $-\frac{\sqrt{2}}{2}<a_{n}<0$.
Therefore, $-1 \leqslant 2 a_{n}^{2}-1<-\frac{1}{2}$, i.e., $-1 \leqslant \cos 2^{n} \theta \leqslant-\frac{1}{2}$.
This contradicts $a_{n}<0$.
From the above derivation, for any positive integer $n$ we have
$-1 \leqslant \cos 2^{n} \theta \leqslant-\frac{1}{4}$.
Thus, $\left|\cos 2^{n} \theta-\frac{1}{2}\right| \geqslant \frac{3}{4}$.
Notice that
$$
\begin{array}{l}
\left|\cos 2 \theta+\frac{1}{2}\right|=\left|2 \cos ^{2} \theta-\frac{1}{2}\right| \\
=2\left|\cos \theta+\frac{1}{2}\right| \cdot\left|\cos \theta-\frac{1}{2}\right| \\
\geqslant \frac{3}{2}\left|\cos \theta+\frac{1}{2}\right| .
\end{array}
$$

Then $\left|\cos \theta+\frac{1}{2}\right| \leqslant \frac{2}{3}\left|\cos 2 \theta+\frac{1}{2}\right|$
$\leqslant\left(\frac{2}{3}\right)^{2}\left|\cos 4 \theta+\frac{1}{2}\right| \leqslant \cdots$
$\leqslant\left(\frac{2}{3}\right)^{n}\left|\cos 2^{n} \theta+\frac{1}{2}\right| \leqslant\left(\frac{2}{3}\right)^{n}$.
As $n \rightarrow+\infty$, $\left(\frac{2}{3}\right)^{n} \rightarrow 0$.
Thus, $\left|\cos \theta+\frac{1}{2}\right|=0$, i.e., $\cos \theta=-\frac{1}{2}$.
Solving gives $\theta=2 k \pi \pm \frac{2 \pi}{3}(k \in \mathbf{Z})$.
On the other hand, when $\theta=2 k \pi \pm \frac{2 \pi}{3}$, for any natural number $n$ we have $\cos 2^{n} \theta=-\frac{1}{2}$, satisfying the condition.
In summary, the set of all real numbers $\theta$ is
$$
\left\{\theta \left\lvert\, \theta=2 k \pi \pm \frac{2 \pi}{3}(k \in \mathbf{Z})\right.\right\} .
$$