(a)(i) The sum to infinity of a geometric progression is given by \(\frac{a}{1-r} = \frac{1}{\cos \theta}\), where \(a = \cos \theta\) and \(r\) is the common ratio.

Rearranging gives \(1 - r = \cos^2 \theta\), so \(r = 1 - \cos^2 \theta = \sin^2 \theta\).

Thus, the second term is \(a \cdot r = \cos \theta \cdot \sin^2 \theta\).

(a)(ii) The sum of the first 12 terms is given by:

\(S_{12} = \frac{a(1-r^{12})}{1-r}\)

Substituting \(a = \cos \frac{\pi}{3} = 0.5\) and \(r = \sin^2 \frac{\pi}{3} = 0.75\), we have:

\(S_{12} = \frac{0.5(1-(0.75)^{12})}{1-0.75}\)

Calculating gives \(S_{12} = 1.937\).

(b) For the arithmetic progression, the common difference \(d = \cos \theta \sin^2 \theta - \cos \theta\).

Substituting \(\theta = \frac{1}{3} \pi\), we find \(d = -\frac{1}{8}\).

The 85th term is given by:

\(a + 84d = \frac{1}{2} + 84 \left(-\frac{1}{8}\right)\)

Calculating gives the 85th term as \(-10\).