Решите уравнение cos^4 x-cos(x+(pi)/(3))cos(x-(pi)/(3))=2sin(x+(pi)/(6))sin(x-(pi)/(6)).

Задача №17623: Уравнение

cos^{4} x - cos (x + \frac{π}{3}) cos (x - \frac{π}{3}) = 2 sin (x + \frac{π}{6}) sin (x - \frac{π}{6}) ⟺

⟺ cos^{4} x - cos (x + \frac{π}{3}) cos (x - \frac{π}{3}) = 2 cos (\frac{π}{3} - x) cos (\frac{2 π}{3} - x) ⟺

⟺ cos^{4} x - cos (x + \frac{π}{3}) cos (x - \frac{π}{3}) = - 2 cos (x - \frac{π}{3}) cos (x + \frac{π}{3}) ⟺

⟺ cos^{4} x + cos (x + \frac{π}{3}) cos (x - \frac{π}{3}) = 0 ⟺ cos^{4} x + \frac{1}{2} (cos 2 x + cos \frac{2 π}{3}) = 0 ⟺

⟺ cos^{4} x + cos^{2} x - \frac{3}{4} = 0 ⟺ (cos^{2} x - \frac{1}{2}) (cos^{2} x + \frac{3}{2}) = 0 ⟺

⟺ cos x = \pm \frac{1}{2} ⟺ x = \frac{π}{4} + \frac{k π}{2}, k \in Z .