№1 (x^2+x+1)^2-3x^2-3x-1=0 №2 (ctgx)^2-(tgx)^2=16cos2x

Решим уравнения:

№1 (x^2 + x + 1)^2 - 3 * x^2 - 3 * x - 1 = 0;  

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

Пусть x^2 + x = t, тогда: 

(t + 1)^2 - 3 * t - 1 = 0; 

t^2 + 2 * t * 1 + 1^2 - 3 * t - 1 = 0; 

t^2 - t + 1 - 1 = 0; 

t^2 - t = 0; 

t * (t - 1) = 0; 

t = 0 и t = 1. 

Тогда: 

{ x^2 + x = 0;  

x^2 + x = 1; 

{ x * (x + 1) = 0; 

x^2 + x - 1 = 0;  

{ x = 0; 

x = 1; 

x₁ = (-1 - √5)/2; 

x₂ = (-1 + √5)/2.

№2 (ctg x)^2 - (tg x)^2 = 16 * cos (2 * x); 

(ctg x)^2 - (tg x)^2 - 16 * cos (2 * x) = 0;  

ctg^2 x - tg^2 x - 16 * (cos^2 x - sin^2 x) = 0; 

ctg^2 x - tg^2 x - 16 * cos^2 x + 16 * sin^2 x = 0; 

cos^2 x/sin^2 x - sin^2 x/cos^2 x - 16 * cos^2 x + 16 * sin^2 x = 0;  

(cos^2 x * cos^2 x - sin^2 x * sin^2 x - 16 * cosa^2 x * cos^2 x * sin^2 x + 16 * sin^2 x* sin^2 x * cos^2 x)/(sin^2 x * cos^2 x) = 0;  

Уравнение имеет корни, если числитель дроби равен 0. 

cos^2 x * cos^2 x - sin^2 x * sin^2 x - 16 * cosa^2 x * cos^2 x * sin^2 x + 16 * sin^2 x* sin^2 x * cos^2 x = 0; 

cos^4 x - sin^4 x - 16 * cos^4 x * sin^2 x + 16 * sin^4 x * cos^2 x = 0; 

cos^2 x - sin^2 x - 16 * sin^2 x * cos^2 x * (cos^2 x - sin^2 x) = 0; 

(cos^2 x - sin^2 x) * (1 - 16 * sin^2 x * cos^2 x) = 0; 

1) 1 - 16 * sin^2 x * cos^2 x = 0; 

(1 - 4 * cos x * sin x) * (1 + cos x * sin x) = 0; 

Нет корней. 

2) cos^2 x - sin^2 x = 0; 

1 - sin^2 x - sin^2 x = 0; 

2 * sin^2 x = 1; 

{ sin x = √2/2; 

sin x = -√2/2; 

{ x = (-1)^n * pi/4 + pi * n; 

x = (-1)^n * (-pi/4) + pi * n.