Prove that \(\dfrac{\sec2\theta-1}{\sec2\theta+1}\equiv\tan^2\theta\), where \(-\dfrac\pi4<\theta<\dfrac\pi4\).[4]
Hence, or otherwise, find the values of \(\theta\) for which \(3(\sec2\theta-1)=\sec2\theta+1\), where \(-\dfrac\pi4<\theta<\dfrac\pi4\).[3]
It is given that the first three terms of a geometric sequence are \(\sec x,\ \operatorname{cosec}2x\) and \(\dfrac12\operatorname{cosec}x\operatorname{cosec}2x\) respectively, where \(0<x<\dfrac\pi2\).
Find the common ratio of the sequence.[2]
Find the range of values of \(x\) for which the geometric series \(\sec x+\operatorname{cosec}2x+\dfrac12\operatorname{cosec}x\operatorname{cosec}2x+\ldots\) converges.[4]
Given that \(x=\arccos\left(\dfrac12\right)\), show that the sum to infinity of the series is \(3+\sqrt3\).[4]