6) sin22x = sin²x;

Решим уравнение $$sin^2(2x) = sin^2x$$

$$sin^2(2x) - sin^2x = 0$$

$$(sin(2x) - sinx)(sin(2x) + sinx) = 0$$

$$sin(2x) - sinx = 0$$ или $$sin(2x) + sinx = 0$$

$$2sin(\frac{2x-x}{2})cos(\frac{2x+x}{2}) = 0$$ или $$2sin(\frac{2x+x}{2})cos(\frac{2x-x}{2}) = 0$$

$$sin(\frac{x}{2})cos(\frac{3x}{2}) = 0$$ или $$sin(\frac{3x}{2})cos(\frac{x}{2}) = 0$$

$$sin(\frac{x}{2}) = 0$$ или $$cos(\frac{3x}{2}) = 0$$ или $$sin(\frac{3x}{2}) = 0$$ или $$cos(\frac{x}{2}) = 0$$

$$\frac{x}{2} = \pi n, n \in Z$$ или $$\frac{3x}{2} = \frac{\pi}{2} + \pi n, n \in Z$$ или $$\frac{3x}{2} = \pi n, n \in Z$$ или $$\frac{x}{2} = \frac{\pi}{2} + \pi n, n \in Z$$

$$x = 2\pi n, n \in Z$$ или $$x = \frac{\pi}{3} + \frac{2\pi}{3} n, n \in Z$$ или $$x = \frac{2\pi}{3} n, n \in Z$$ или $$x = \pi + 2\pi n, n \in Z$$

Ответ: $$x = 2\pi n, n \in Z$$ или $$x = \frac{\pi}{3} + \frac{2\pi}{3} n, n \in Z$$ или $$x = \frac{2\pi}{3} n, n \in Z$$ или $$x = \pi + 2\pi n, n \in Z$$