Для удобства заменим tg(x) на t:

tg(3x) - t / (1 + tg(3x) * t = -1.

tg(3x) - t = - (1 + tg(3x) * t).

tg(3x) - t = -1 - tg(3x) * t.

tg(3x) + tg(3x) * t = t - 1.

tg(3x) * (1 + t) = t - 1.

tg(3x) = (t - 1) / (1 + t).

tg(3x) = t / (1 + t).

tg(3x) = sin(3x) / cos(3x) = t / (1 + t).

sin(3x) / cos(3x) = t / (1 + t).

sin(3x)(1 + t) = t * cos(3x).

sin(3x) + sin(3x) t = t cos(3x).

sin(3x) - t * cos(3x) = 0.

sin(3x) = t * cos(3x).

(sin(3x))^2 = (t * cos(3x))^2.

sin^2(3x) = t^2 * cos^2(3x).

1 - cos^2(3x) = t^2 * cos^2(3x).

1 = t^2 * cos^2(3x) + cos^2(3x).

1 = cos^2(3x)(t^2 + 1).

cos^2(3x) = 1 / (t^2 + 1).

cos(3x) = ± sqrt(1 / (t^2 + 1)).

3x = ± arccos(sqrt(1 / (t^2 + 1))).

x = ± arccos(sqrt(1 / (t^2 + 1))) / 3.

Итак, x = ± arccos(sqrt(1 / (tg^2(x) + 1))) / 3.