1) Cosx - cos3x = 3sin^2x 2) 3tg^2x - 8cos^2x + 1 = 0

Вычислим корни уравнений.

1) Cos x - cos (3 * x) = 3 * sin^2 x;

Упростим уравнение.

-2 * sin ((x - 3 * x) / 2) * sin ((x + 3 * x) / 2) = 3 * sin^2 x;

-2 * sin (-2 * x/2) * sin (4 * x/2) = 3 * sin^2 x;

-2 * sin (-x) * sin (2 * x) = 3 * sin^2 x;

2 * sin x * sin (2 * x) = 3 * sin^2 x;

3 * sin^2 x - 2 * sin x * sin (2 * x) = 0;

3 * sin^2 x - 2 * sin x * 2 * sin x * cos x = 0;

Применим свойства степеней.

3 * sin^2 x - 4 * sin^2 x * cos x = 0;

sin^2 x * (3 - 4 * cos x) = 0;

{ sin^2 x = 0;

3 - 4 * cos x = 0;

{ sin x = 0;

4 * cos x = 3;

{ x = пи * n, n ∈ Z;

x = + -arccos (3/4) + 2 * пи * n, n ∈ Z.

2) 3 * tg^2 x - 8 * cos^2 x + 1 = 0;

3 * sin^2 x/cos^2 x - 8 * cos^2 x + 1 = 0;

3 * sin^2 x - 5 * cos^4 x + cos^2 x = 0;

3 * (1 - cos^2 x) - 5 * cos^4 x + cos^2 x = 0;

-5 * cos^4 x + cos^2 x - 3 * cos^2 x + 3 = 0;

5 * cos^4 x + 2 * cos^2 x - 3 = 0;

D = 4 - 4 * 5 * (-3) = 4 + 20 * 3 = 64 = 8^2;

cos^2 x = (-2 + 8) / (2 * 5) = 6/10 = 3/5;

cos^2 x = (-2 - 8) / 10 = - 10/10 = - 1;

1) cos^2 x = - 1;

Нет корней.

2) cos^2 x = 0.6;

cos x = + -√0.6;

{ cos x = + √0.6;

cos x = - √0.6;

{ x = + -arccos (√0.6) + 2 * пи * n, n ∈ Z;

x = + -arccos (-√0.6) + 2 * пи * n, n ∈ Z.