AMC 12 2024a Problem 23

What is the value of

$$\tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {3\pi}{16}~ + ~ \tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {5\pi}{16} ~+~\tan^2 \frac {3\pi}{16} \cdot \tan^2 \frac {7\pi}{16} ~+~ \tan^2 \frac {5\pi}{16} \cdot \tan^2 \frac {7\pi}{16}?$$

$\textbf{(A) } 28 \textbf{(B) } 68 \textbf{(C) } 70 \textbf{(D) } 72 \textbf{(E) } 84$