To solve the equation \(\tan x \tan 2x = 1\), we start by using the identity for \(\tan 2x\):

\(\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}\)

Substitute this into the equation:

\(\tan x \cdot \frac{2 \tan x}{1 - \tan^2 x} = 1\)

Simplify the equation:

\(\frac{2 \tan^2 x}{1 - \tan^2 x} = 1\)

Cross-multiply to clear the fraction:

\(2 \tan^2 x = 1 - \tan^2 x\)

Rearrange the terms:

\(3 \tan^2 x = 1\)

Divide by 3:

\(\tan^2 x = \frac{1}{3}\)

Take the square root:

\(\tan x = \pm \frac{1}{\sqrt{3}}\)

Thus, \(x = 30^\circ, 150^\circ\) in the given interval \(0^\circ < x < 180^\circ\).