Use the formula cos2θ = 2cos^{2}θ – 1 to transform the equation

cos2θ + cosθ = 0 => 2cos^{2}θ + cosθ – 1 = 0.

Factor the equation as (2cosθ – 1)(cosθ + 1) = 0.

There are two series of solutions: 2cosθ – 1 = 0; cosθ = 1/2; θ = +– cos^{–1}(1/2) + 2πk;

θ = +– π/3 + 2πk, where k is any integer (k = 0, 1, 2, 3, …, –1, –2, –3, ...)

and cosθ + 1 = 0; cosθ = – 1; θ = π + 2πk, where k is any integer (k = 0, 1, 2, 3, …, –1, –2, –3, ...)

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