To solve the equation 4cos(x)sin(x) - √3 = 0, we can use the trigonometric identity for the product of sin and cos:

cos(2x) = 2cos(x)sin(x)

Let's rewrite the equation in terms of cos(2x):

4cos(x)sin(x) - √3 = 0
2 * 2cos(x)sin(x) - √3 = 0
2cos(2x) - √3 = 0

Now, solve for cos(2x):

2cos(2x) = √3
cos(2x) = √3 / 2

The values of cos(2x) are ±√3/2. To find the values of x, we need to find the solutions for 2x:

2x = ±π/6 + 2πn, where n is an integer.

Finally, divide by 2 to find the values of x:

x = ±π/12 + πn, where n is an integer.

Therefore, the solutions to the equation 4cos(x)sin(x) - √3 = 0 are x = π/12 + πn and x = -π/12 + πn, where n is an integer.