How to solve $\sqrt{\tan x}=1$ without squaring? Problem:
$$\sqrt{\tan x}=1$$
Solution (with squaring):
$$\sqrt{\tan x}=1$$
$$\tan x=1$$
$$x=n\pi+\frac{\pi}{4}\tag{1}$$
1. $(i)$ contains extraneous roots. How do I filter them out?
2. How do I solve for $x$ without squaring?


$$\sqrt{\tan x}=1\\\tan x=1\\x=n\pi+\frac{\pi}{4}$$

You have shown that $$\sqrt{\tan x}=1\implies\tan x=1\implies x=n\pi+\frac{\pi}{4}$$ (the given equation's implicit condition that $\tan x\ge0$ justifies $\left(\sqrt{\tan x}\right)^2=\tan x$ in the first step).
But notice also that $$x=n\pi+\frac{\pi}{4}\implies\tan x=1\implies\sqrt{\tan x}=1$$ (taking principal square root in the last step).
Therefore, in this example, all three lines are equivalent to one another; so, no extraneous root has been created. Hence, squaring does not necessarily create extraneous roots.
