Express an integral in terms of another integral.

You can apply Integration by Parts to the original integral $\displaystyle \int_0^{\pi/6}\tan^{12}(2x)\sec(2x)\,dx$.

If we set $$u=\tan^{11}(2x) \quad \text{and} \quad dv=\tan(2x)\sec(2x)\,dx,$$ then $$du=22\tan^{10}(2x)\sec^2(2x)\,dx \quad \text{and} \quad v=\frac{1}{2}\sec(2x),$$ and the integral turns into $$\int_0^{\pi/6}\tan^{12}(2x)\sec(2x)\,dx=\left.\frac{1}{2}\tan^{11}(2x)\sec(2x)\right|_0^{\pi/6}-11\int_0^{\pi/6}\tan^{10}(2x)\sec^3(2x)\,dx.$$ And to the last integral you can apply what you did but backwards.