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6Cos²x - 7Cosx - 5 = 0
Сделаем замену Cosx = m , - 1 ≤ Cosx ≤ 1
6m² - 7m - 5 = 0
D = (- 7)² - 4 * 6 * (- 5) = 49 + 120 = 169 = 13²
$m_{1} = \frac{7-13}{12} =- \frac{1}{2}$
$m_{2} = \frac{7+13}{12}= \frac{5}{3}=1 \frac{2}{3} \ \textgreater \ 1$
m₂ - не подходит
$Cosx=- \frac{1}{2}\\\\ x_{1}= \frac{2 \pi }{3} +2 \pi n \\\\ x_{2}=- \frac{2 \pi }{3}+2 \pi n\\\\\\- \pi \leq \frac{2 \pi }{3}+2 \pi n \leq 2 \pi \\\\-1 \leq \frac{2}{3} +2n \leq 2\\\\-1 \frac{2}{3} \leq 2n \leq 1 \frac{1}{3} \\\\- \frac{5}{6} \leq n \leq \frac{2}{3} \\\\n = 0 , x = \frac{2 \pi }{3}+2 \pi *0= \frac{2 \pi }{3} \\\\\\- \pi \leq - \frac{2 \pi }{3}+2 \pi n \leq 2 \pi \\\\-1 \leq - \frac{2}{3}+2n \leq 2\\\\- \frac{1}{3} \leq 2n \leq 2 \frac{2}{3}$
$- \frac{1}{6} \leq n \leq 1 \frac{1}{3}\\\\n=0,x=- \frac{2 \pi }{3} +2 \pi *0=- \frac{2 \pi }{3} \\\\n=1,x= - \frac{2 \pi }{3} +2 \pi *1= \frac{4 \pi }{3}$

Ответ: $- \frac{2 \pi }{3} ; \frac{2 \pi }{3} ; \frac{4 \pi }{3}$