1) cos5x + sin3x = 0
cos5x = -sin3x
cos^2(5x) = sin^2(3x)
1 - sin^2(5x) = sin^2(3x)
1 - sin^2(5x) = 1 - cos^2(3x)
sin^2(5x) = cos^2(3x)
sin^2(5x) = 1 - sin^2(3x)
sin^2(5x) = 1 - cos^2(5x)
sin^2(5x) + cos^2(5x) = 1
1 = 1

2) sin2x - cosx = 2sinx - 1
sin2x - 2sinx = cosx - 1
2sinx cosx - 2sinx = cosx - 1
2sinx cosx - cosx = 2sinx - 1
cosx(2sinx - 1) = 2sinx - 1
cosx = (2sinx - 1) / (2sinx - 1)
cosx = 1

3) sin2x + sin4x + cosx = 0
2sinxcosx + 2sin2x cos2x + cosx = 0
2sinxcosx + 2sinxcos2x + cosx = 0
2sinxcosx(1 + 2cosx) + cosx = 0
cosx(2sinx + 1) = 0

Итак, решениями уравнений будут:
1) x = kπ, где k - целое число
2) x = (2k + 1)π/2, где k - целое число
3) x = π/2 или x = -π/6.