How to find a line cuts a circle internally?

Question

I have a circle $S:x^2+y^2=4$ and points $A(2\cos \theta,2\sin \theta)$ and $B(2,0)$

Let $L$ be the tangent to $S$ at $A$.

$C$ and $D$ are distinct point on $L$ such that $CA=AD=1$

How can I find the range of $\theta$ such that both $BC$ and $BD$ cut $S$ internally?

My attempt:

I can easily obtain $L:(\cos\theta)x+(\sin \theta)y-2=0$

But I am stuck in the following.

Can I get some clues?

Answer 1

Construct the tangent to the circle at $B$. Let $D', D''$ be the points on this tangent that are one unit from $B$. From each of the latter points construct the other tangent to the circle; these lines intersect the circle at $A'$ and $A''$ respectively such that $|A'D'|=|A''D''|=1$. This gives the bounding conditions from which you can get your range for $\theta$.