Length of a taut cord passing over a circular pipe, giving an equation in an angle solved numerically

The diagram shows the cross-section of a circular pipe with centre \(O\) and radius \(0.4\text{ m}\). The points \(A\) and \(B\) lie on a horizontal line through \(O\), with \(OA=OB\). A taut cord \(ACDB\) of length \(2\text{ m}\) runs from \(A\) to the pipe, around the arc \(CD\) over the top of the pipe, and then to \(B\). The straight sections \(AC\) and \(DB\) are tangents to the circle at \(C\) and \(D\), and \(\angle OAC=\angle OBD=\theta\) radians.

1. Explain why \(\angle COD=2\theta\), and hence show that \(\theta+\cot\theta=\frac52\).[5]
2. Find the value of \(\theta\), correct to four decimal places.[2]