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Double angle identity with a common factor and the solutions of a related trigonometric equation in radians

Double angle identity with a common factor and the solutions of a related trigonometric equation in radians

8 marks
  1. Show that \(\sin2x-\cos2x+1=2\sin x\,(\sin x+\cos x)\).[2]
  2. Using part (a), or otherwise, find all values of \(x\), where \(0\leq x\leq2\pi\), that satisfy \(\sin2x-\cos2x+1=\sin x+\cos x\).[6]

Solution:

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Answer:(a) \(2\sin x\cos x+2\sin^2x=2\sin x\,(\sin x+\cos x)\) (b) \(x=\dfrac{\pi}{6},\ \dfrac{3\pi}{4},\ \dfrac{5\pi}{6},\ \dfrac{7\pi}{4}\)

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