(a) The common difference \(d\) is given by the difference between the second term and the first term:

\(d = -\frac{\tan^2 \theta}{\cos^2 \theta} - \frac{1}{\cos^2 \theta}\)

\(= \frac{-\sin^2 \theta - \cos^2 \theta}{\cos^4 \theta}\)

\(= \frac{-1}{\cos^4 \theta}\)

Thus, the common difference is \(-\frac{1}{\cos^4 \theta}\).

(b) The 13th term \(u_{13}\) is given by:

\(u_{13} = a + 12d\)

Where \(a = \frac{1}{\cos^2 \theta}\) and \(d = -\frac{1}{\cos^4 \theta}\).

When \(\theta = \frac{1}{6} \pi\), \(\cos \theta = \frac{\sqrt{3}}{2}\).

\(a = \frac{1}{\left(\frac{\sqrt{3}}{2}\right)^2} = \frac{4}{3}\)

\(d = -\frac{1}{\left(\frac{\sqrt{3}}{2}\right)^4} = -\frac{16}{9}\)

\(u_{13} = \frac{4}{3} + 12 \left(-\frac{16}{9}\right)\)

\(= \frac{4}{3} - \frac{192}{9}\)

\(= \frac{4}{3} - \frac{64}{3}\)

\(= -20\)

Thus, the exact value of the 13th term is \(-20\).