The problem is $\arctan(\cos u) = ?$ I think for this I need to find an angle answer in relation to $u$. So I think of a triangle where
$\cos u = x$ and so $\tan v = x$
since we need the angle $v$ such that the tan of it is $\;= \cos u = x$
I then divided $u$ by $v$ to get
$$\frac{u}{v} = \frac{\arccos(x)}{\arctan(x)}$$
An x value that $\arccos$ and $\arctan$ have in common is $-1$.
$$\frac{u}{v} = \frac{\arccos(-1)}{\arctan(-1)} = \frac{\pi}{\frac{-\pi}{4}} = -4$$ and because $\frac{u}{v} = -4$ then $v = \frac{-u}{4}$ This is the relationship and I believe this is the answer $$\arctan(\cos u) = \frac{-u}{4}$$ I am not sure if this is correct or not, are these steps valid? Also is there a way to do this without plugging in values like $x = -1$ and keeping it purely with variables or something like that?