A curve is parametrically represented as

\[\begin{cases} x = \cos t+\ln \left(\tan\frac{t}{2}\right) \\ y = \sin t, \end{cases} \]

where \(t\) is a parameter.

Find the length of tangent to the curve at the point where its \(x\)-coordinate is equal to its \(y\)-coordinate.

The length of tangent is defined as the distance between the point of contact with the curve and the point where the tangent meets the \(x\)-axis.

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