Separate the variables:

\(\frac{dx}{x+1} = \frac{\cos 2θ}{\sin 2θ} \, dθ\)

Integrate both sides:

\(\int \frac{dx}{x+1} = \int \frac{\cos 2θ}{\sin 2θ} \, dθ\)

\(\ln |x+1| = \frac{1}{2} \ln |\sin 2θ| + C\)

Use the initial condition \(θ = \frac{1}{12}π, x = 0\):

\(\ln |0+1| = \frac{1}{2} \ln |\sin \frac{1}{6}π| + C\)

\(0 = \frac{1}{2} \ln \frac{1}{2} + C\)

\(C = -\frac{1}{2} \ln \frac{1}{2}\)

Substitute back:

\(\ln |x+1| = \frac{1}{2} \ln |\sin 2θ| - \frac{1}{2} \ln \frac{1}{2}\)

\(\ln |x+1| = \frac{1}{2} \ln \left( \frac{\sin 2θ}{\frac{1}{2}} \right)\)

\(\ln |x+1| = \frac{1}{2} \ln (2 \sin 2θ)\)

\(x+1 = \sqrt{2 \sin 2θ}\)

\(x = \sqrt{2 \sin 2θ} - 1\)