4. In $\triangle A B C$, if $\sin A=2 \sin C$, and the three sides $a, b, c$ form a geometric sequence, then the value of $\cos A$ is . $\qquad$
$$
\begin{array}{l}
a=2 c \Rightarrow b^{2}=a c=2 c^{2} \Rightarrow b=\sqrt{2} c \Rightarrow \cos A=\frac{b^{2}+c^{2}-a^{2}}{2 b c} \\
=\frac{2 c^{2}+c^{2}-4 c^{2}}{2 \sqrt{2} c^{2}}=-\frac{\sqrt{2}}{4} .
\end{array}
$$