Help in understanding this velocity triangle of a propeller
There is an obvious error in this paper. Just look at the final formula, and remember that for small angles $\tan(\phi) \sim \phi$. The formula says that if you increase $\phi$, $v_2$ decreases. Of course, the opposite happens.
By (trigonometric) definition,
$$\tan(\phi) = \frac{v_2}{v_{propeller}}$$
Therefore,
$$v_2 = v_{propeller} \cdot \tan(\phi)$$
The rest follows; the result is similar but you have multiplication instead of the fraction.
P.S. You can get a hint that this is a low quality paper by the fact that they obliviously confuse upper- and lowercase $v$ (on the picture and in the formulae). In math/science, case matters! Sometimes even the script matters.