2 sin x + 2 cos x =0| : cos x

Divide by cos x (cos x ≠ 0 in any actual solution): 2 sin x + 2 cos x = 0 => 2(sin x / cos x) + 2 = 0 => 2 tan x + 2 = 0 => tan x = -1

General solution: x = -π/4 + kπ, k ∈ ℤ

(Equivalent form: x = 3π/4 + kπ, k ∈ ℤ)

Note: cos x = 0 does not satisfy the original equation, so division is safe.