AMC 12 2000 Problem 17

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A circle centered at $O$ has radius $1$ and contains the point $A$. The segment $AB$ is tangent to the circle at $A$ and $\angle AOB = \theta$. If point $C$ lies on $\overline{OA}$ and $\overline{BC}$ bisects $\angle ABO$, then $OC =$

$\textbf {(A)}\ \sec^2 \theta - \tan \theta \textbf {(B)}\ \frac 12 \textbf {(C)}\ \frac{\cos^2 \theta}{1 + \sin \theta} \textbf {(D)}\ \frac{1}{1+\sin\theta} \textbf {(E)}\ \frac{\sin \theta}{\cos^2 \theta}$