To solve this equation, we will first simplify it by using the properties of logarithms.

Given equation: (2cos^2x - sinx - 1)log0.5(-0.5cosx) = 0

Using the property of logarithms that log_a(b) = log_c(b) / log_c(a), we can rewrite the equation as:

log0.5(-0.5cosx) = 0

Now, we can rewrite the logarithmic equation in exponential form:

0.5^0 = -0.5cosx

1 = -0.5cosx

Solving for cosx:

cosx = -1/0.5

cosx = -2

Since the cosine function varies between -1 and 1, there is no real solution for cosx = -2. Therefore, the original equation does not have a solution.