cos3x-sinx=√3 (cosx-sin3x)

1. Раскроем скобки:

cos3x - sinx = √3 (cosx - sin3x); cos3x - sinx = √3cosx - √3sin3x; √3sin3x + cos3x - (√3cosx + sinx) = 0.

2. Разделим на 2:

√3/2 * sin3x + 1/2 * cos3x - (√3/2 * cosx + 1/2 * sinx) = 0; cos (π/6) * sin3x + sin (π/6) * cos3x - (sin (π/3) * cosx + cos (π/3) * sinx) = 0; sin (3x + π/6) - sin (x + π/3) = 0.

3. Разность синусов:

sina - sinb = 2sin ((a - b) / 2) * cos ((a + b) / 2); 2sin{ ((3x + π/6) - (x + π/3)) / 2} * cos{ ((3x + π/6) + (x + π/3)) / 2} = 0; sin{ (3x + π/6 - x - π/3) / 2} * cos{ (3x + π/6 + x + π/3) / 2} = 0; sin (x - π/12) * cos (2x + π/4) = 0; [sin (x - π/12) = 0;

[cos (2x + π/4) = 0; [x - π/12 = πk, k ∈ Z;

[2x + π/4 = π/2 + πk, k ∈ Z; [x = π/12 + πk, k ∈ Z;

[2x = π/4 + πk, k ∈ Z; [x = π/12 + πk, k ∈ Z;

[x = π/8 + πk/2, k ∈ Z.

Ответ: π/12 + πk; π/8 + πk/2, k ∈ Z.