Cos^2 (5x) + 7sin^2 (5x) = 8 sin5x cos5x

1. Разделим обе части уравнения на cos^2 (5x):

cos^2 (5x) + 7sin^2 (5x) = 8sin5x * cos5x; cos^2 (5x) / cos^2 (5x) + 7sin^2 (5x) / cos^2 (5x) = 8sin5x * cos5x/cos^2 (5x); 1 + 7tg^2 (5x) = 8tg5x; 7tg^2 (5x) - 8tg5x + 1 = 0.

2. Введем новую переменную:

tg5x = t.

Тогда:

7t^2 - 8t + 1 = 0; D/4 = 4^2 - 7 * 1 = 16 - 7 = 9; t = (4 ± √9) / 7 = (4 ± 3) / 7;

1) t = (4 - 3) / 7 = 1/7;

tg5x = 1/7; 5x = arctg (1/7) + πk, k ∈ Z; x = 1/5 * arctg (1/7) + πk/5, k ∈ Z;

2) t = (4 + 3) / 7 = 7/7 = 1;

tg5x = 1; 5x = π/4 + πk, k ∈ Z; x = π/20 + πk/5, k ∈ Z.

Ответ: 1/5 * arctg (1/7) + πk/5; π/20 + πk/5, k ∈ Z.