7. Find all values of $\alpha$ such that all terms in the sequence $\cos \alpha, \cos 2 \alpha, \cdots, \cos 2^{n} \alpha \cdots$ are negative.
7. $\cos \alpha$ must be less than $-\frac{1}{4}$, and for all $n \in \mathrm{N}$, $\cos 2^{n} \alpha > \frac{3}{4}$.
Also, $\left|\cos \alpha+\frac{1}{2}\right|=\frac{2}{3}\left|\cos \alpha+\frac{1}{2}\right|$, which leads to the recursive formula
$$
\begin{array}{l}
\left|\cos \alpha+\frac{1}{2}\right|<\frac{2}{3}\left|\cos 2 \alpha+\frac{1}{2}\right|<\left(\frac{2}{3}\right)^{2}\left|\cos 2^{2} \alpha+\frac{1}{2}\right| \\
<\cdots<\left(\frac{2}{3}\right)^{n}\left|\cos 2^{n} \alpha+\frac{1}{2}\right|<\left(\frac{2}{3}\right)^{n+1}\left|\cos 2^{n+1} \alpha+\frac{1}{2}\right|<\cdots,
\end{array}
$$

which means $\left|\cos \alpha+\frac{1}{2}\right|$ can be less than any positive number, hence $\left|\cos \alpha+\frac{1}{2}\right|=0$, i.e., $\alpha=2 k \pi \pm \frac{2 \pi}{3} k \in Z$.