1) sin^4x-cos^4x=sin2x-1/22) sin^4x+cos^4x=cos^2 2x

Решите уравнения:

1) sin⁴x - cos⁴x = sin (2x) - 1/2;

2) sin⁴x + cos⁴x = cos² (2x);

Решение.

1) sin⁴x - cos⁴x = sin (2x) - 1/2;

(sin²x + cos²x) (sin²x - cos²x) = sin (2x) - 1/2;

- cos (2x) = sin (2x) - 1/2;

cos (2x) + sin (2x) = 1/2;

sin (2x) * √2/2 + cos (2x) * √2/2 = 1/2 * √2/2;

sin (2x + π/4) = √2/4;

2x + π/4 = arcsin (√2/4) + 2πk; π - arcsin (√2/4) + 2πk;

2x = arcsin (√2/4) - π/4 + 2πk; 3π/4 - arcsin (√2/4) + 2πk;

2x = π/4 ± (arcsin (√2/4) - π/2) + 2πk;

x = π/8 ± (1/2arcsin (√2/4) - π/4) + πk, k ∈ Z.

2) sin⁴x + cos⁴x = cos² (2x);

sin⁴x + cos⁴x = (cos²x - sin²x) ²;

sin⁴x + cos⁴x = cos⁴x - 2 * cos²x * sin²x + sin⁴x;

2 * cos²x * sin²x = 0;

4 * cos²x * sin²x = 0;

(2 * sinx * cosx) ² = 0;

sin² (2x) = 0;

sin (2x) = 0;

2x = πk;

x = πk/2, k ∈ Z.