1) 2cos5x + √3=0
2cos5x = -√3
cos5x = -√3/2
5x = ±2π/3 + 2kπ
x = ±2π/15 + 2kπ, where k is an integer.

2) 8sinx + 5 = 2cos2x
8sinx + 5 = 2(1 - 2sin²x)
8sinx + 5 = 2 - 4sin²x
4sin²x + 8sinx + 3 = 0
(2sinx + 1)(2sinx + 3) = 0
sinx = -1/2 or sinx = -3/2 (no real solution)

3) cos²(x/3) - 5sin(x/3)cos(x/3) = 3
cos²(x/3) - 5sin(x/3)cos(x/3) - 3 = 0
(cos(x/3) - 3)(cos(x/3) + 1) = 0
cos(x/3) = 3 or cos(x/3) = -1 (no real solution)

4) (2sinx - 1)sinx = sin2x - cosx
2sin²x - sinx = 2sinxcosx - cosx
2sin²x - 3sinx + 1 = 0
sinx = (3 ± √5)/4
x = arcsin((3 ± √5)/4) + 2kπ or x = π - arcsin((3 ± √5)/4) + 2kπ, where k is an integer.

5) cos(π+x) - sin(π/2 +x) - sin2x = 0
-sin(x) - cos(x) - sin2x = 0
-sin(x) - cos(x) - 2sinxcosx = 0
-sin(x)(1 + 2cosx) - cos(x) = 0
sin(x)(2cos(x) + 1) + cos(x) = 0
tan(x) = -2

6) 5sin2x - 2cosx = 0
5(2sinxcosx) - 2cosx = 0
10sinxcosx - 2cosx = 0
2(5sinx - 1)cosx = 0
cosx = 0 or sinx = 1/5 (no real solution)

7) cos2x - cos6x = 7sin²x
2cos²x - 1 - (32cos³x - 48cosx) = 7 - 7cos²x
2cos²x - 1 - 32cos³x + 48cosx = 7 - 7cos²x
2cos²x + 7cos²x - 32cos³x - 48cosx - 8 = 0
cosx = -4/3 or cosx = 2
x = arccos(-4/3) + 2kπ or x = arccos(2) + 2kπ, where k is an integer.

8) √2sin10x + sin2x = cos2x
sin(10x) = √2 - 1
10x = arcsin(√2 - 1) + 2kπ or 10x = π - arcsin(√2 - 1) + 2kπ, where k is an integer.