Let $x$ be the optimal distance from the sign, $\alpha$ be the subtended angle of the sign and $\beta$ be the angle formed by the ground and the ray from the optimal viewpoint to the bottom of the sign. Then $\tan\beta=10/x$ and $\tan(\alpha+\beta)=20/x$. Using the formula for the tangent of difference, you can express $\tan((\alpha+\beta)-\beta)=\tan\alpha$ through $x$. Since $\tan$ is a monotonic function on $[0,\pi/2)$, the maximum point of $\tan(\alpha(x))$ is also the maximum point of $\alpha(x)$.