To solve the equation cos(3x - π/6) - sin(4x) = 0, we can use the trigonometric identity cos(a - b) = cos(a)cos(b) + sin(a)sin(b).

Using this identity, we can rewrite the equation as:

cos(3x)cos(π/6) + sin(3x)sin(π/6) - sin(4x) = 0

Since cos(π/6) = √3/2 and sin(π/6) = 1/2, we have:

(√3/2)cos(3x) + (1/2)sin(3x) - sin(4x) = 0

Multiplying through by 2 to clear the denominators, we get:

√3cos(3x) + sin(3x) - 2sin(4x) = 0

Now, let's use the trigonometric identity sin(2θ) = 2sin(θ)cos(θ) to rewrite sin(4x) as:

sin(4x) = 2sin(2x)cos(2x)

Substituting this back into the equation, we have:

√3cos(3x) + sin(3x) - 2(2sin(2x)cos(2x)) = 0

Simplifying further, we get:

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

Now, let's use the trigonometric identity cos(2θ) = 1 - 2sin^2(θ) to rewrite cos(2x) as:

cos(2x) = 1 - 2sin^2(x)

Substituting this back into the equation, we have:

√3cos(3x) + sin(3x) - 4sin(2x)(1 - 2sin^2(x)) = 0

Expanding and simplifying, we get:

√3cos(3x) + sin(3x) - 4sin(2x) + 8sin^3(x) = 0

This equation is now in terms of sin(x) and cos(x). To solve for x, we can use numerical methods or approximations.