Имеем уравнение:
(1 - tg(x))(1 + sin^2(x)) = 1 + tg(x)

Раскрываем скобки:
(1 - tg(x))(1 + sin^2(x)) = 1 + tg(x)
1 + sin^2(x) - tg(x) - sin^2(x) tg(x) = 1 + tg(x)
sin^2(x) - tg(x) - sin^2(x) tg(x) = tg(x)

sin^2(x) - tg(x) - sin^2(x) * tg(x) - tg(x) = 0
sin^2(x) - tg(x)(1 + sin^2(x)) - tg(x) = 0
sin^2(x) - tg(x)(1 + sin^2(x) + 1) = 0
sin^2(x) - tg(x)(sin^2(x) + 2) = 0
sin^2(x) - 2tg(x) - tg(x)sin^2(x) = 0

Теперь подставим tg(x) = sin(x) / cos(x) и sin^2(x) = 1 - cos^2(x):

1 - cos^2(x) - 2(sin(x) / cos(x)) - (sin(x) / cos(x))(1 - cos^2(x)) = 0
1 - cos^2(x) - 2sin(x) / cos(x) - (sin(x) / cos(x) - cos^2(x)*sin(x) / cos(x)) = 0
1 - cos^2(x) - 2sin(x) / cos(x) - sin(x) / cos(x) + cos(x)sin(x) = 0
1 - cos^2(x) - 3sin(x) / cos(x) + cos(x)sin(x) = 0

1 - cos^2(x) = sin^2(x), поэтому:

sin^2(x) - 3sin(x) / cos(x) + cos(x)sin(x) = 0
sin(x)(sin(x) - cos(x)) - 3sin(x) / cos(x) = 0
(sin(x) - 3)sin(x) / cos(x) = 0

Отсюда получаем два решения с учетом промежутка [-180; 180]:
1) sin(x) = 0 => x = 0 или x = 180
2) sin(x) = 3*cos(x) => x = arctg(3)

Сумма корней: 0 + 180 = 180
Ответ: 180.