\section*{Problem 1 - 121231}

Determine all real numbers \(x\) that satisfy the inequality \(0<x<\frac{\pi}{2}\) and the equation

\[
\tan x + \cot x = 4
\]

(Computation of the numerical values as decimal fractions is not required.)
}

Using the transformation

\[
\tan x + \cot x = \frac{\sin^2 x + \cos^2 x}{\sin x \cos x} = \frac{2}{\sin (2x)}
\]

the original equation can also be simply written as

\[
\sin (2x) = \frac{1}{2}
\]

with the two solutions

\[
x = \frac{\pi}{12} \quad \text{or} \quad x = \frac{5\pi}{12}
\]

in the interval \(\left(0, \frac{\pi}{2}\right)\).