1) sinx + cosx = 0
sinx = -cosx
sinx/cosx = -1
tanx = -1
x = 3π/4 + kπ, где k - целое число

2) cos(3π/2 + x) - 5cosx = 0
-sinx - 5cosx = 0
sinx = -5cosx
sinx/cosx = -5
tanx = -5
x = arctan(-5) + kπ, где k - целое число

3) 5sin3x - 2cos3x = 0
5sin3x = 2cos3x
5tan3x = 2
tan3x = 2/5
3x = arctan(2/5) + kπ, где k - целое число
x = (arctan(2/5) + kπ)/3

4) 2sin^2x + 3sinxcosx - 2cos^2x = 0
2sin^2x + 3sinx - 5sinx - 2 = 0
(2sinx - 1)(sinx + 2) = 0
sinx = 1/2 или sinx = -2
x = π/6 + 2πk, где k - целое число или x = asin(-2), где а - любое действительное число

5) 6cos^2x - 2sin2x = 1
6(1 - sin^2x) - 4sinxcosx = 1
6 - 6sin^2x - 4sinx(2sinx*cosx) = 1
6 - 6sin^2x - 8sin^2x = 1
-14sin^2x = -5
sin^2x = 5/14
x = ±arcsin(√(5/14)) + kπ, где k - целое число

6) 2cos^2x + 5sin2x - 4 = 0
2(1 - sin^2x) + 52sinxcosx - 4 = 0
2 - 2sin^2x + 10sinxcosx - 4 = 0
-2sin^2x + 5sinxcosx - 1 = 0
-2sin^2x + sin2x - 4sin2x + 4 = 0
-sin^2x - 4sin2x + 4 = 0
sin^2x + 4sin2x - 4 = 0
(sin x)^2 + 42*sinxcosx - 4 = 0
(sin x + 2cosx)^2 - 8cos^2x - 4 = 0
(√2sin(x + π/4))^2 - 8cos^2x - 4 = 0
(√2sin(x + π/4) - 2√2cosx)(√2sin(x + π/4) + 2√2cosx) = 0
√2(sin(x + π/4) - 2cosx)(sin(x + π/4) + 2cosx) = 0

√2(sin(x + π/4) - 2cosx) = 0 или sin(x + π/4) + 2cosx = 0
sin(x + π/4) = 2cosx/√2
sin(x + π/4) = √2*cosx
x + π/4 = arccos(1/sqrt(3)) + 2kπ или x + π/4 = -arccos(1/sqrt(3)) + 2kπ, где k - целое число

sin(x + π/4) + 2cosx = 0
sinxcos(π/4) + cosxsin(π/4) + 2cosx = 0
sin(x + π/4) + 2cosx = 0
sinx + cosx + 2cosx = 0
cosx = -sinx
cosx/sinx = -1
tanx = -1
x = arctan(-1) + kπ, где k - целое число