How do I simplify this sum of arccosines? After trying to solve a geometry problem, represented by the following image

I've arrived at this expression:
$\alpha=\arccos\left(\frac{d+r \cos \left(\varphi+\frac{\vartheta}2\right)}{\sqrt{d^2+r^2+2 dr\cos \left(\varphi+\frac{\vartheta}2\right)}}\right)+\arccos\left(\frac{d+r\cos \left(\varphi-\frac{\vartheta}2\right)}{\sqrt{d^2+r^2+2dr\cos\left(\varphi-\frac{\vartheta}2\right)}}\right)$
Is there a way to simplify such expression?

$$\arccos \beta +\arccos \gamma=\arccos \left(\beta\cdot\gamma-\sqrt{1-\beta^2} \cdot \sqrt{1-\gamma^2}\right)$$
On Wikipedia there are also these identities:
$$\arccos x_1+\arccos x_2=
\begin{cases}
\arccos\left(x_1x_2-\sqrt{1-x_1^2}\sqrt{1-x_2^2}\right)&
x_1+x_2\ge0\\
2\pi-\arccos\left(x_1x_2-\sqrt{1-x_1^2}\sqrt{1-x_2^2}\right)&
x_1+x_2<0
\end{cases}
$$
$$\arccos x_1-\arccos x_2=
\begin{cases}
-\arccos\left(x_1x_2+\sqrt{1-x_1^2}\sqrt{1-x_2^2}\right)&
x_1\ge x_2\\
\arccos\left(x_1x_2+\sqrt{1-x_1^2}\sqrt{1-x_2^2}\right)&
x_1<x_2
\end{cases}
$$
