Sine Synesthesia

The expression we want to maximize can also be written, by substituting $t = \sin(x)$ , as $-2t^2+t+1$ . Analyzing the vertex of this parabola, we see that the expression holds its maximum value for $t = \sin(x) = \frac{1}{4}$ . Because $\sin(2x) = 2 \sin(x) \cos(x)$ , we have that $\cos(x)$ is positive, which leads to $\cos(x) = \frac{\sqrt{15}}{4}$ and $\cot(x) = \sqrt{15}$ . By the double cotangent formula, $\cot(2x) = \frac{7}{\sqrt{15}}$ and thus our answer is $\boxed{7.}$