To solve the equation cot(x) + tan(x) = 2sin(2x), we can use trigonometric identities.

First, let's simplify the left side of the equation using the identities cot(x) = 1/tan(x) and sin(2x) = 2sin(x)cos(x):

1/tan(x) + tan(x) = 2(2sin(x)cos(x))

Next, let's find a common denominator for the left side:

(1 + tan^2(x))/tan(x) = 4sin(x)cos(x)

Using the identity 1 + tan^2(x) = sec^2(x), we can rewrite the equation as:

sec^2(x)/tan(x) = 4sin(x)cos(x)

Now, let's simplify the right side of the equation:

sec^2(x)/tan(x) = 4sin(x)cos(x)
sec^2(x)/(sin(x)/cos(x)) = 4sin(x)cos(x)
sec^2(x)cos(x)/sin(x) = 4sin(x)cos(x)

Canceling out the common factors of sin(x) and cos(x), we get:

sec^2(x) = 4

Taking the square root of both sides, we have:

sec(x) = ±2

Since sec(x) is the reciprocal of cos(x), we can rewrite this as:

cos(x) = ±1/2

To find the values of x that satisfy this equation, we can look at the unit circle or use inverse trigonometric functions.

For cos(x) = 1/2, we have two solutions:

x = π/3 + 2πn, where n is an integer
x = 5π/3 + 2πn, where n is an integer

For cos(x) = -1/2, we also have two solutions:

x = 2π/3 + 2πn, where n is an integer
x = 4π/3 + 2πn, where n is an integer

Therefore, the solutions to the equation cot(x) + tan(x) = 2sin(2x) are:

x = π/3 + 2πn, where n is an integer
x = 2π/3 + 2πn, where n is an integer
x = 4π/3 + 2πn, where n is an integer
x = 5π/3 + 2πn, where n is an integer