9 The first term of a progression is \(\cos \theta\), where \(0 < \theta < \frac { 1 } { 2 } \pi\).
- For the case where the progression is geometric, the sum to infinity is \(\frac { 1 } { \cos \theta }\).
- Show that the second term is \(\cos \theta \sin ^ { 2 } \theta\).
- Find the sum of the first 12 terms when \(\theta = \frac { 1 } { 3 } \pi\), giving your answer correct to 4 significant figures.
- For the case where the progression is arithmetic, the first two terms are again \(\cos \theta\) and \(\cos \theta \sin ^ { 2 } \theta\) respectively.
Find the 85 th term when \(\theta = \frac { 1 } { 3 } \pi\).
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The diagram shows a sector \(A B C\) which is part of a circle of radius \(a\). The points \(D\) and \(E\) lie on \(A B\) and \(A C\) respectively and are such that \(A D = A E = k a\), where \(k < 1\). The line \(D E\) divides the sector into two regions which are equal in area.