Ratio of Sum of Squared Magnitudes of Vectors Involving Angle Fractions

Given the vectors $\vec{a}_k = (\tan \theta_k)\hat{i} + \hat{j}$ and $\vec{b}_k = \hat{i} - (\cot \theta_k)\hat{j}$, their squared magnitudes are given by $|\vec{a}_k|^2 = \tan^2 \theta_k + 1 = \sec^2 \theta_k$ and $|\vec{b}_k|^2 = 1 + \cot^2 \theta_k = \csc^2 \theta_k$.

For the angles $\theta_k = \frac{2^{k-1}\pi}{2^n + 1}$ ($k = 1, 2, \dots, n$), using trigonometric identities for sums of secant and cosecant squares over these specific angle distributions, the sum of squared magnitudes satisfies the exact ratio relationship:

$\frac{\sum_{k=1}^n |\vec{a}_k|^2}{\sum_{k=1}^n |\vec{b}_k|^2} = \frac{\sum_{k=1}^n \sec^2 \theta_k}{\sum_{k=1}^n \csc^2 \theta_k} = 3$

Thus, the value of the expression is $3$.