6sin²x + 4sinxcosx = 1
6sin²x + 4sinxcosx = sin²x + cos²x
5sin²x + 4sinxcosx - cos²x = 0
Разделим на cos²x.
5tg²x + 4tgx - 1 = 0
Пусть t = tgx.
5t² + 4t - 1 = 0
D = 16 + 4•5 = 36 = 6².
t1 = (-4 + 6)/10 = 2/10 = 1/5
t2 = (-4 - 6)/10 = -10/10 = -1
Обратная замена:
tgx = 1/5
x = arctg(1/5) + πn, n ∈ Z.
tgx = -1
x = -π/4 + πn, n ∈ Z.
Ответ: х = arctg(1/5) + πn, n ∈ Z; x = -π/4 + πn, n ∈ Z.