Sooo many TANS

$\tan^2 \frac \pi{16} + \tan^2 \frac {2\pi}{16} + \tan^2 \frac {3\pi}{16} + \tan^2 \frac {4\pi}{16} + \tan^2 \frac {5\pi}{16} + \tan^2 \frac {6\pi}{16} + \tan^2 \frac {7\pi}{16} \\ = \color{#3D99F6}{\tan^2 \frac \pi{16}} + \color{#D61F06}{\tan^2 \frac {\pi}{8}} + \color{#20A900}{\tan^2 \frac {3\pi}{16}} + \tan^2 \frac {\pi}{4} + \color{#20A900}{\tan^2 \frac {5\pi}{16}} + \color{#D61F06}{\tan^2 \frac {3\pi}{8}} + \color{#3D99F6}{\tan^2 \frac {7\pi}{16}} \\ = \color{#3D99F6}{\tan^2 \frac \pi{16} + \tan^2 \frac {7\pi}{16}} + \color{#D61F06}{\tan^2 \frac {\pi}{8} + \tan^2 \frac {3\pi}{8}} + \color{#20A900}{\tan^2 \frac {3\pi}{16} + \tan^2 \frac {5\pi}{16}} + \tan^2 \frac {\pi}{4} \quad \quad \small \color{#3D99F6}{\text{See Note}} \\ = \color{#3D99F6}{\dfrac 4{\sin^2 \frac \pi 8}-2} + \color{#D61F06}{\dfrac 4{\sin^2 \frac \pi 4}-2} + \color{#20A900}{\dfrac 4{\sin^2 \frac {3\pi}8}-2} + 1 \quad \quad \small \color{#3D99F6}{\text{Note that }\sin \frac \pi 4 = \frac 1{\sqrt 2}} \\ = \color{#3D99F6}{\dfrac 4{\frac 12(1-\cos \frac \pi 4)}-2} + \color{#D61F06}{8-2} + \color{#20A900}{\dfrac 4{\frac 12(1-\cos \frac {3\pi} 4)}-2} + 1 \\ = 8\left(\dfrac 1{1- \frac 1{\sqrt 2}} + \dfrac 1{1+ \frac 1{\sqrt 2}} \right)+ 3 \\ = 8(4) + 3 = \boxed{35}$

---

$\color{#3D99F6}{\text{Note:}}$

$\begin{aligned} \tan^2 \frac \pi{16} + \tan^2 \frac {7\pi}{16} & = \tan^2 \frac \pi{16} + \cot^2 \frac \pi{16} \\ & = \left(\tan \frac \pi{16} + \cot \frac \pi{16}\right)^2 - 2\tan \frac \pi{16} \cot \frac \pi{16} \\ & = \left(\frac {\sin \frac \pi{16}}{\cos \frac \pi{16}} + \frac {\cos \frac \pi{16}}{\sin \frac \pi{16}} \right)^2 - 2 \frac {\sin \frac \pi{16}}{\cos \frac \pi{16}} \cdot \frac {\cos \frac \pi{16}}{\sin \frac \pi{16}} \\ & = \left(\frac {\sin^2 \frac \pi{16} + \cos^2 \frac \pi{16}}{\sin \frac \pi{16}\cos^2 \frac \pi{16}} \right)^2 - 2 \\ & = \left(\frac 2{\sin \frac \pi{8}} \right)^2 - 2 \\ & = \frac 4{\sin^2 \frac \pi{8}} - 2 \end{aligned}$