For context, I am trying to show that a certain orthogonal projection operator is not directionally differentiable at a point in direction $d$. The set in question is the convex hull of the set $S = \{(\cos x_n,\sin x_n) : n \geq 1 \} \cup \{(0,0),(1,0)\}$, where $(x_n)$ satisfies $x_1 = \frac{\pi}{2}$, and $x_n$ decreases monotonically to $0$ after that.
I am told that the projection onto this set of the point $\left(2, \sin x_n + (2-\cos x_n)\tan\left(\frac{x_n + x_{n-1}}{2}\right)\right)$ is $(\cos x_n, \sin x_n)$ but I am not sure how to prove this. What I have tried so far is working with the convex constrained optimality conditions and the gradient of the norm to obtain that this is equivalent to proving the inequality $$\tan\left(\frac{x_n + x_{n-1}}{2}\right)(u_2 - \sin x_n) \leq \cos x_n - u_1$$ for all $u = (u_1,u_2) \in \text{conv}(S)$.
However I have been unable to prove this inequality, and maybe there is a better way to do it. Any help is much appreciated.
Let $p_n = (\cos x_n, \sin x_n)$.
Orange polygon is the convex hull of $S$.
Point $C$ is constructed in the following way: it is the intersection of the line $y=2$ and line, that is perpendicular to the segment $p_{n-1}p_n$ and passes through the point $p_n$. So by construction $p_n$ is the projection of $C$ on the segment $p_{n-1}p_n$.
Calculating coordinates of $C$ we find, that $C = \left(2, \sin x_n + (2-\cos x_n)\tan\left(\frac{x_n + x_{n-1}}{2}\right)\right)$.