To solve the equation tan(4x) - cot(x) = 0, we can rewrite cot(x) as 1/tan(x):

tan(4x) - 1/tan(x) = 0

Next, we can multiply both sides of the equation by tan(x) to eliminate the denominators:

tan(4x) * tan(x) - 1 = 0

Using the trigonometric identity tan(a) * tan(b) = (sin(a) * sin(b)) / (cos(a) * cos(b)), we can simplify the equation further:

(sin(4x) * sin(x)) / (cos(4x) * cos(x)) - 1 = 0

Now, let's multiply both sides of the equation by (cos(4x) * cos(x)) to eliminate the denominators:

sin(4x) * sin(x) - (cos(4x) * cos(x)) = 0

Using the trigonometric identity sin(a) * sin(b) - cos(a) * cos(b) = cos(a + b), we can rewrite the equation as:

cos(4x + x) = 0

Simplifying further:

cos(5x) = 0

To find the values of x that satisfy this equation, we need to find the values of 5x for which cos(5x) = 0.

Since cos(5x) = 0 when 5x = π/2 + nπ, where n is an integer, we can solve for x:

5x = π/2 + nπ

x = (π/2 + nπ)/5, where n is an integer.