Question 5 - A-Level Maths

Exam Board	OCR
Module	H240/03 (Pure Mathematics and Mechanics)
Year	2019
Session	June
Topic	Reciprocal Trig & Identities

Prove that \(( \cot \theta + \operatorname { cosec } \theta ) ^ { 2 } = \frac { 1 + \cos \theta } { 1 - \cos \theta }\).
Hence solve, for \(0 < \theta < 2 \pi , 3 ( \cot \theta + \operatorname { cosec } \theta ) ^ { 2 } = 2 \sec \theta\).
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The diagram shows part of the curve \(y = \frac { 2 x - 1 } { ( 2 x + 3 ) ( x + 1 ) ^ { 2 } }\).
Find the exact area of the shaded region, giving your answer in the form \(p + q \ln r\), where \(p\) and \(q\) are positive integers and \(r\) is a positive rational number.