Решение
3tgx-4ctgx=5
3tgx - 4/tgx - 5 = 0
3tg²x - 5tgx - 4 = 0
tgx = t
3t² - 5t - 4 = 0
D = 25 + 4*3*4 = 73
t₁ = (5 - √73)/6
t₂ = (5 + √73)/6
1) tgx = (5 - √73)/6
x₁ = arctg(5 - √73)/6 + πk, k∈Z
2) tgx = (5 + √73)/6
x₂ = arctg(5 + √73)/6 + πn, n∈Z
Ответ: x₁ = arctg(5 - √73)/6 + πk, k∈Z ; x₂ = arctg(5 + √73)/6 + πn, n∈Z