6) 2 cos²x = 1 + 2 sin²x

Решение: $$2 cos^2x = 1 + 2 sin^2x$$ $$2 cos^2x - 2 sin^2x = 1$$ $$2(cos^2x - sin^2x) = 1$$ $$2 cos 2x = 1$$ $$cos 2x = \frac{1}{2}$$ $$2x = \pm \arccos(\frac{1}{2}) + 2\pi k, k \in \mathbb{Z}$$ $$2x = \pm \frac{\pi}{3} + 2\pi k, k \in \mathbb{Z}$$ $$x = \pm \frac{\pi}{6} + \pi k, k \in \mathbb{Z}$$ Ответ: $$x = \pm \frac{\pi}{6} + \pi k, k \in \mathbb{Z}$$