$\begin{array}{l}\text { 6. } \cos 18^{\circ}+2 \cos 36^{\circ}+\cdots+20 \cos 360^{\circ} \\ =\end{array}$
6. 10 .

Let $z=\sum_{k=1}^{20} k \mathrm{e}^{\mathrm{i} 18 k^{\circ}}$. Then $\mathrm{e}^{\mathrm{i} 18^{\circ}} z=\sum_{k=1}^{20} k \mathrm{e}^{\mathrm{i} 18(k+1)^{\circ}}$.
Subtracting the two equations gives
$$
\begin{array}{l}
\left(1-\mathrm{e}^{\mathrm{i} 18^{\circ}}\right) z=\sum_{k=1}^{20} \mathrm{e}^{\mathrm{i} 18 k^{\circ}}-20 \mathrm{e}^{\mathrm{i} 18^{\circ}} \\
\Rightarrow z=\frac{-20 \mathrm{e}^{\mathrm{i} 18^{\circ}}}{1-\mathrm{e}^{\mathrm{i} 18^{\circ}}} .
\end{array}
$$

Therefore, the original expression $=\operatorname{Re} z=10$.