Question 82
Let $w$, $x$, $y$, and $z$ be positive real numbers such that
$$
\begin{aligned}
0 & \neq \cos w \cos x \cos y \cos z \\
2 \pi & = w + x + y + z \\
3 \tan w & = k(1 + \sec w) \\
4 \tan x & = k(1 + \sec x) \\
5 \tan y & = k(1 + \sec y) \\
6 \tan z & = k(1 + \sec z)
\end{aligned}
$$
(Here $\sec t$ denotes $\frac{1}{\cos t}$ when $\cos t \neq 0$.) Find $k$.
$$
\begin{aligned}
0 & \neq \cos w \cos x \cos y \cos z \\
2 \pi & = w + x + y + z \\
3 \tan w & = k(1 + \sec w) \\
4 \tan x & = k(1 + \sec x) \\
5 \tan y & = k(1 + \sec y) \\
6 \tan z & = k(1 + \sec z)
\end{aligned}
$$
(Here $\sec t$ denotes $\frac{1}{\cos t}$ when $\cos t \neq 0$.) Find $k$.