3tg (x/2) + ctgx=5/sinx

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