This has probably already been answered but I can't find any solution using only high-school trigonometry. I'm not a native English speaker, so I probably lack the proper wording to find what I was looking for. Anyway:
In a plan $P$, I have two secant lines $d_1$ and $d_2$ making an angle of $\alpha$. I have a second plan $P'$ containing $d_2$ and making an angle of $\beta$ with $P$ (see image below).
Using only $\alpha$, $\beta$ and the basic trigonometric functions, I would like to express the angle $\alpha '$ made by the projection of $d_1$ and $d_2$ onto the $P'$ plan.
I came myself to the conclusion that $\alpha ' = atan(tan \beta . cos \alpha)$, but I'm not confident at all in the path I followed to reach that. It appears I was wrong. See answer below.