2ctg (x-1) + 3tg (x-1) = 5

Решение

2ctg (x-1) + 3tg (x-1) = 5

2 * (1/tg (x-1) + 3tg (x-1) = 5 умножим на tg (x-1) ≠ 0

3tg ² (x-1) - 5tg (x-1) + 2 = 0

tg (x-1) = t

3t² - 5t + 2 = 0

D = 25 - 4*3*2 = 1

t₁ = (5 - 1) / 6

t₁ = 2/3

t₂ = (5 + 1) / 6

t₂ = 1

1) tg (x-1) = 2/3

x - 1 = arctg (2/3) + πk, k ∈ Z

x₁ = arctg (2/3) + 1 + πk, k ∈ Z

2) tg (x-1) = 1

x-1 = π/4 + πn, n ∈ Z

x₂ = π/4 + 1 + πn, n ∈ Z

Ответ: x₁ = arctg (2/3) + 1 + πk, k ∈ Z; x₂ = π/4 + 1 + πn, n ∈ Z