Question

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

Answer 1

Решение
2ctg(x-1)+3tg(x-1)=5 
2*(1/tg(x-1)+3tg(x-1)=5 умножим на tg(x-1) ≠ 0 
3tg²(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