cos^2x - sin^2x = -0.5
(cosx - sinx)(cosx + sinx) = -0.5
(cosx - sinx)(cosx + sinx) = -1/2
This is a valid trigonometric identity.

sin(П - x) - cos(П/2 + x) = √3
sin(П)cos(x) - cos(П)sin(x) - cos(П/2)sin(x) -sin(П/2)cos(x) = √3
0 - sin(x) + 0 - cos(x) = √3
-sin(x) - cos(x) = √3
This is not a valid trigonometric identity.

cos(П + x) = sin(П/2)
cos(П)cos(x) - sin(П)sin(x) = sin(П/2)
-1cos(x) + 0sin(x) = 1
-cos(x) = 1
This is not true.

3cos(x) - sin(2x) = 0
3cos(x) - 2sin(x)cos(x) = 0
3cos(x) - 2sin(x)cos(x) = 0
cos(x)(3 - 2sin(x)) = 0
cos(x) = 0 or 3 - 2sin(x) = 0
cos(x) = 0 or sin(x) = 3/2
Therefore, there is no solution for sin(x) = 3/2

2sin(x)cos(x) = 0.5
sin(2x) = 0.5
2x = π/6 or 2x = 5π/6
x = π/12 or x = 5π/12

cos^2(x) = 1 + sin^2(x)
1 - sin^2(x) = 1 + sin^2(x)
1 - 1 = sin^2(x) + sin^2(x)
0 = 2sin^2(x)
sin^2(x) = 0
sin(x) = 0
This is valid.

9sin(4x) = 0
sin(4x) = 0
4x = πn where n is an integer
x = (πn)/4 where n is an integer.