Cosx*tgx+cosx+tgx+1=0

Решение

Cosx*tgx+cosx+tgx+1=0

Cosx * (sinx/cosx) + cosx + sinx/cosx + 1 = 0 умножим на сosx ≠ 0

sinx*cosx + cos²x + sinx + cosx = 0

cosx (sinx + cosx) + (sinx + cosx) = 0

(sinx + cosx) * (cosx + 1) = 0

1) sinx + cosx = 0 делим на cosx ≠ 0

tgx + 1 = 0

tgx = 1

x₁ = π/4 + πk, k ∈Z

2) cosx + 1 = 0

cosx = - 1

x₂ = π + 2πn, n∈Z

Ответ: x₁ = π/4 + πk, k ∈Z; x₂ = π + 2πn, n∈Z