Tg (x/2) + ctg x = sin (x/2)/ cos (x/2) + cos x / sin x =
= (sin (x/2)·sin x + cos x·cos (x/2)) / (cos (x/2)·sin x ) =
= cos (x - x/2) / (cos (x/2)·sin x ) =
= cos (x/2) / (cos (x/2)·sin x ) = 1 / sin x
2tg (x/2) + 1 / sin x = 5 / sin x
2tg (x/2) = 5 / sin x - 1 / sin x
2tg (x/2) = 4/ sin x
tg (x/2) = 2/ sin x
tg (x/2) =√((1 - cos x) / (1 + cos x)) = (1 - cos x)/ sin x
(1 - cos x)/ sin x = 2/ sin x
(1 - cos x - 2)/ sin x = 0
(cos x + 1)/ sin x = 0
cos x = -1 при этом sin x≠0
Получается, нет корней