# What is the solution of cos2x - cos x = 0 text{in the interval} [0, 2pi ) ?

Question
What is the solution of $$\cos2x - \cos x = 0\ \text{in the interval}\ [0, 2\pi )$$ ?

2021-02-14
Let use the trigoniometric identity $$\cos2x = 2\cos^{2}x - 1:$$
$$2\cos^{2}x - 1 - \cos x = 0$$
Factorize: $$(\cos x - 1) (2\cos x + 1) = 0$$
Use the zero product property: $$\cos x - 1 = 0\ or\ 2\cos x + 1 = 0$$
Solve both equations to consine: $$\cos x = 1\ or\ \cos x = -\frac{1}{2}$$
$$\cos x = 1$$ when $$x = 0$$
$$x = -\frac{1}{2}$$ when $$x = \pi - \pi/3 = (2\pi)/3\ or\ x = \pi + \pi/3 = (4\pi)/3$$
Aswer: $$x = 0, (\frac{2\pi}{3}), (4\pi)/3$$

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