Answer
Step 3 (perform the division) is wrong, as we cannot simplify $\frac{\tan2\theta}{2}$ into $\tan\theta$, leading to wrong results in step 4.
Work Step by Step
Solve $\tan2\theta=2$ over the interval $[0,2\pi)$
We examine each step:
1) Original equation: $$\tan2\theta=2$$
This is obviously correct.
2) Divide by 2
$$\frac{\tan2\theta}{2}=\frac{2}{2}$$
This is probably weird but still correct.
3) Perform the division
$$\tan\theta=1$$
Now this step is wrong. Divide the whole $\tan2\theta$ by $2$ would not make it into $\tan\theta$. Instead, it would stay the same as $\frac{\tan2\theta}{2}$. Only $\tan(\frac{2\theta}{2})$ could change to $\tan\theta$
4) Definition of inverse tangent
$$\theta=\frac{\pi}{4}\hspace{1cm}\text{or}\hspace{1cm}\theta=\frac{5\pi}{4}$$
Step 3 was carried out wrongly, leading to wrong results in step 4, though the application of inverse tangent definition is correct.
