Let a_k =( tan θ_k) i + j and b_k = i -( cot θ_k) j, where θ_k=(2^k-1 π)/(2^n+1), for some n ∈ N, n>5. Then the value of (Σ_k=1^n| a_k |^2)/(Σ_k=1^n|…

Let $\displaystyle \overrightarrow{a_k}=\left(\tan \theta_k\right) \hat{i}+\hat{j}$ and $\displaystyle \overrightarrow{b_k}=\hat{i}-\left(\cot \theta_k\right) \hat{j}$, where $\displaystyle \theta_k=\frac{2^{k-1} \pi}{2^n+1}$, for some $\displaystyle n \in \mathbb{N}, n>5$. Then the value of $\displaystyle \frac{\sum_{k=1}^n\left|\overrightarrow{a_k}\right|^2}{\sum_{k=1}^n\left|\overrightarrow{b_k}\right|^2}$ is $\displaystyle \_\_\_\_$.