3sin^2x-4sinxcosx+5cos^x=2

3sin^2x - 4sinxcosx + 5cos^2x = 2;
3sin^2х - 4sinxcosx + 5cos^2x - 2 = 0;
3sin^2х - 4sinxcosx + 5cos^2x - 2·(sin^2x+cos^2x) = 0;
3sin^2х - 4sinxcosx + 5cos^2x - 2sin^2x - 2cos^2x=0
Sin^2x-4sinxcosx+3cos^2x=0.
Разделим на cos^2x:

cosx ≠ 0; x ≠ п/2 + пn, n∈Z;

tg^2x - 4tgx + 3 = 0;
Сделаем замену переменной:
tgx = t;
t^2 -4t + 3 = 0;
D = 16 - 12 = 4;
t1 = (4 + 2)/2 = 3;
t2 = (4 - 2)/2 = 1;
tgx = 1;
x = arctg1 + пk;k∈Z;
x = п/4 + пk;k∈Z;
tgx = 3;
x = arctg3 + пn;n∈Z.
Ответ: x = п/4 + пk; k∈Z; x = arctg3 + пn; n∈Z.