1. 2cos4x - √2 = 0;
2cos4x = √2;
cos4x = √2/2;
4x = ±arccos(√2/2) + 2πn, n∈Z;
4x = ±π/4 + 2πn, n∈Z;
x = ±π/16 + πn/2, n∈Z;
2. cos(arcsin(-1/2) + arctg√3 + arccos(1/2)) = cos(-π/6 + π/3 + π/3) = cos(π/2) = 0.
Отже, cos(arcsin(-1/2) + arctg√3 + arccos(1/2)) = 0. Тотожність доведено.
3.
а) 2sin²x - 2cos²x = 1;
-2(cos²x - sin²x) = 1;
-2cos2x = 1;
cos2x = -1/2;
2x = ±arccos(-1/2) + 2πn, n∈Z;
2x = ±2π/3 + 2πn, n∈Z;
x = ±π/3 + πn, n∈Z.
б) ОДЗ: sin4x ≠ 0
З врахуванням ОДЗ маємо:
4x=π/2 + πn, n∈Z;
x=π/8 + πn/2, n∈Z;