Where do I look at this from? (2)

Consider a circle $S_{1}$ ,

$x^{2}+y^{2}+1-2x(\cos\theta - \sin\theta)-2y(\cos\theta + \sin\theta)=0$

Now, consider another circle $S_{2}$ , centred at $P=(-\cos\theta - \sin\theta, \cos\theta - \sin\theta)$ such that $S_{1}$ internally touches $S_{2}$ .

Draw a pair of tangents $T_{1}$ and $T_{2}$ from $P$ to $S_{1}$ . Let these tangents meet the circle $S_{2}$ at points $A$ and $B$ as shown. From a point $R$ on $S_{2}$ , draw chords $RA$ and $RB$ with lengths $l_{1} , l_{2}$ , respectively to $S_{2}$ .

Then, there exists a certain $\alpha$ such that one of

$\large \left| \cos^{-1}\frac{l_{1}}{6} + \cos^{-1}\frac{l_{2}}{6}\right| \quad \text{ or } \quad \left|\cos^{-1}\frac{l_{1}}{6}-\cos^{-1}\frac{l_{2}}{6} \right|$

is equal to $\alpha$ regardless of the value of $R$ . Find the value of this constant $\alpha$ .

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