Let $S$ be the set of all real values of $x$ with $0 < x < \frac{\pi}{2}$ such that $\sin x$, $\cos x$, and $\tan x$ form the side lengths (in some order) of a right triangle.  Compute the sum of $\tan^2 x$ over all $x$ in $S$.
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $S$ be the set of all real values of $x$ with $0 < x < \frac{\pi}{2}$ such that $\sin x$, $\cos x$, and $\tan x$ form the side lengths (in some order) of a right triangle.  Compute the sum of $\tan^2 x$ over all $x$ in $S$.
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\sqrt{2}
Let $S$ be the set of all real values of $x$ with $0 < x < \frac{\pi}{2}$ such that $\sin x$, $\cos x$, and $\tan x$ form the side lengths (in some order) of a right triangle.  Compute the sum of $\tan^2 x$ over all $x$ in $S$.
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