If $\displaystyle \theta_1$ and $\displaystyle \alpha$ are two real numbers such that $\displaystyle \frac{\cot^4 \theta}{\csc^4 \theta}\sec^2 \alpha$, $\displaystyle \frac{1}{\csc 150^{\circ}}$, $\displaystyle \frac{\tan^4 \theta}{\sec^4 \theta}\csc^2 \alpha$ are in Arithmetic Progression, prove that:

$\displaystyle \frac{\sec^{2n} \alpha}{\sec^{2n + 2} \theta}$, $\displaystyle \frac{1}{\csc 150^{\circ}}$, $\displaystyle \frac{\csc^{2n} \alpha}{\csc^{2n + 2} \theta}$ are also in A.P. for all $\displaystyle n\in \mathbb{N}$