Сначала найдем tgA и ctgA:

sinA + cosA = 1/3
(sinA)^2 + 2sinAcosA + (cosA)^2 = 1/9
sin^2 + 2sinAcosA + cos^2 = 1/9
1 + 2sinAcosA = 1/9
2sinAcosA = 1/9 - 1
2sinAcosA = -8/9
sinAcosA = -4/9

tgA = sinA/cosA = -4/9 / (1 - sin^2) = -4/9 / cos^2 = -4 / (9cos)
ctgA = cosA/sinA = - 9/4

Подставляем значения в начальное уравнение:

1 + 2(-4/9) + (-9/4) = 1 + (-8/9) - 9/4 = 1 - 8/9 - 9/4 = 36/36 - 32/36 - 81/36 = -77/36

Поэтому 1 + 2/tgA + ctgA = 1 + 2/(-4/9) - 9/4 = 1 - 18/4 - 9/4 = -26/4 = -13/2.