To solve the equation cos(2x) + 0.5/(cos(x)sin(x)) = 0, we will first simplify the expression by using trigonometric identities.

Recall that cos(2x) = cos^2(x) - sin^2(x) and that cos(x)sin(x) = 0.5sin(2x).

Therefore, the equation becomes:

cos^2(x) - sin^2(x) + 0.5/(0.5sin(2x)) = 0
cos^2(x) - sin^2(x) + 1/sin(2x) = 0

Now, we can rewrite sin^2(x) as 1 - cos^2(x) and sin(2x) as 2sin(x)cos(x):

cos^2(x) - (1 - cos^2(x)) + 1/(2sin(x)cos(x)) = 0
cos^2(x) - 1 + cos^2(x) + 1/(2sin(x)cos(x)) = 0
2cos^2(x) - 1 + 1/(2sin(x)cos(x)) = 0

Therefore, the simplified equation is:

2cos^2(x) - 1 + 1/(2sin(x)cos(x)) = 0

This is a trigonometric equation that can be solved further by using trigonometric identities and algebraic manipulation.