Length of a line in an isosceles triangle. (mind boggling )

Question

In an isosceles $\triangle ABC$, side $AB$ and $AC$ are equal in length. There exists a point $D$ on the side $AB$. $\angle BAC$ is $\theta$. The side $AD$ is $2$ units smaller than $AC$. What is the generalized formula to calculate the side $CD$?

Answer 1

You can draw a picture to convince yourself that the length of $BD$ does not depend on $\theta$. You will need to be given more information.

Answer 2

This is a simple application of the cosine rule to the triangle $ACD$

$$CD^2=(a-2)^2+a^2-2a(a-2)\cos\theta$$