2sinx=1+cosx sin^2x-5sinxcosx+4cos^2x=0

1.

2sinx = 1 + cosx; 2 * 2sin (x/2) cos (x/2) = 1 + 2cos^2 (x/2) - 1; 2 * 2sin (x/2) cos (x/2) - 2cos^2 (x/2) = 0; cos (x/2) (2sin (x/2) - cos (x/2)) = 0; [cos (x/2) = 0;

[2sin (x/2) - cos (x/2) = 0; [cos (x/2) = 0;

[tg (x/2) = 1/2; [x/2 = π/2 + πk, k ∈ Z;

[x/2 = arctg (1/2) + πk, k ∈ Z; [x = π + 2πk, k ∈ Z;

[x = 2arctg (1/2) + 2πk, k ∈ Z.

2.

sin^2x - 5sinxcosx + 4cos^2x = 0; tg^2x - 5tgx + 4 = 0; D = 5^2 - 4 * 4 = 9 = 3^2;

1) tgx = (5 - 3) / 2 = 1;

x = π/4 + πk, k ∈ Z.

2) tgx = (5 + 3) / 2 = 4;

x = arctg (4) + πk, k ∈ Z.