III) tg²x - tgx - 12 = 0

Решение:

Пусть \( y = \text{tg} x \). Уравнение: \( y^2 - y - 12 = 0 \).

Дискриминант \( D = (-1)^2 - 4 \cdot 1 \cdot (-12) = 1 + 48 = 49 \). \( \sqrt{D} = 7 \).

\( y_1 = (1 + 7)/2 = 4 \), \( y_2 = (1 - 7)/2 = -3 \).

1) \( \text{tg} x = 4 \) \( \implies x = \text{arctg}(4) + \pi n \).

2) \( \text{tg} x = -3 \) \( \implies x = \text{arctg}(-3) + \pi k = -\text{arctg}(3) + \pi k \).

Ответ: \( x = \text{arctg}(4) + \pi n \), \( x = -\text{arctg}(3) + \pi k \).