Tg²x - 2sin²x = 0
sin²x/cos²x - 2sin²x = 0
ОДЗ:
cosx ≠ 0
x ≠ π/2 + πm, m ∈ Z
sin²x - 2sin²xcos²x = 0
sin²x(1 - 2cos²x) = 0
Произведение множителей равно нулю, если хотя бы один из множителей равен нулю:
1) sinx = 0
x = πn, n ∈ Z - данное решение не уд. ОДЗ
2) 1 - 2cos²x = 0
-(2cos²x - 1) = 0
-cos2x = 0
cos2x = 0
2x = π/2 + πk, k ∈ Z
x = π/4 + πk/2, k ∈ Z
-3π/4 ≤ πn ≤ 2π, n ∈ Z
-0,75 ≤ n ≤ 2, n ∈ Z
n = 0; 1; 2.
x₁ = -π;
x₂ = 0
x₃ = π
-3π/4 ≤ π/4 + πk/2 ≤ 2π, k ∈ Z
-3 ≤ 1 + 2k ≤ 8, k ∈ Z
k = -2; -1; 0; 1; 2; 3
x₄ = π/4 - π = -3π/4
x₅ = π/4 - π/2 = -π/4
x₆ = π/4
x₇ = π/4 + π/2 = 3π/4
x₈ = π/4 + π = 5π/4
x₉ = π/4 + 3π/2 = 7π/4
Ответ: x = πn, n ∈ Z; π/4 + πk/2, k ∈ Z; -π; 0; π; -3π/4; -π/4; π/4; 3π/4; 5π/4; 7π/4.