To solve the equation cos(3x) - sin(5x) = √3(cos(5x) - sin(3x), we can start by simplifying both sides of the equation.

First, let's expand the trigonometric functions using the angle addition and subtraction formulas:

cos(3x) - sin(5x) = √3(cos(5x) - sin(3x))
cos(3x) - sin(5x) = √3(cos(5x) - cos(π/2 - 3x))
cos(3x) - sin(5x) = √3(cos(5x) - cos(π/2)cos(3x) + sin(π/2)sin(3x))
cos(3x) - sin(5x) = √3(cos(5x) - 0 + 1*sin(3x))
cos(3x) - sin(5x) = √3(cos(5x) + sin(3x))

Now, let's simplify the equation:

cos(3x) - sin(5x) = √3(cos(5x) + sin(3x))
cos(3x) - sin(5x) = √3cos(5x) + √3sin(3x)

Next, let's group the terms with cos(3x) and sin(5x) together:

cos(3x) - √3cos(5x) = sin(5x) + √3sin(3x)

Now, we can use the trigonometric identity cos(a) - cos(b) = -2sin((a+b)/2)sin((a-b)/2) to simplify the left side of the equation:

cos(3x) - √3cos(5x) = -2sin((3x+5x)/2)sin((3x-5x)/2)
cos(3x) - √3cos(5x) = -2sin(4x)sin(-x)
cos(3x) - √3cos(5x) = 2sin(4x)sin(x)

Now, we can use the trigonometric identity sin(a) + sin(b) = 2sin((a+b)/2)cos((a-b)/2) to simplify the right side of the equation:

sin(5x) + √3sin(3x) = 2sin((5x+3x)/2)cos((5x-3x)/2)
sin(5x) + √3sin(3x) = 2sin(4x)cos(x)

Now, we have:

cos(3x) - √3cos(5x) = 2sin(4x)sin(x)
sin(5x) + √3sin(3x) = 2sin(4x)cos(x)

Since the left side of the first equation is equal to the right side of the second equation, and the left side of the second equation is equal to the right side of the first equation, we can set them equal to each other:

cos(3x) - √3cos(5x) = sin(5x) + √3sin(3x)

Now, we can solve this equation for x. However, it is a nonlinear equation and does not have a simple algebraic solution. We would need to use numerical methods or graphing techniques to find the approximate values of x that satisfy the equation.