To solve this equation, we can use the trigonometric identities:

cot(x) = 1/tan(x) and tan(x + 45°) = (tan(x) + 1)/(1 - tan(x))

Using these identities, we can rewrite the equation as:

tan(x + 30°) = 1/tan(2x + 45°)

Now, let's simplify the equation:

tan(x + 30°) = (tan(2x + 45°))^-1

tan(x + 30°) = (1 - tan(2x + 45°))/(tan(2x + 45°))

Now, let's substitute tan(2x + 45°) with its equivalent expression:

tan(x + 30°) = (1 - (tan(x) + 1)/(1 - tan(x)))/(tan(x) + 1)/(1 - tan(x))

tan(x + 30°) = (1 - tan(x) - 1)/(tan(x) + 1)/(1 - tan(x))

tan(x + 30°) = -tan(x)/(tan(x) + 1)/(1 - tan(x))

Now, let's simplify the equation further:

tan(x + 30°) = -tan(x)/(tan^2(x) - 1)

tan(x + 30°) = -tan(x)/(-sec^2(x))

tan(x + 30°) = tan(x)/sec^2(x)

tan(x + 30°) = tan(x)cos^2(x)

Now, let's solve for x:

tan(x)cos^2(x) = tan(x + 30°)

cos^2(x) = 1/cos^2(x + 30°)

cos^2(x) = sec^2(x + 30°)

cos(x) = sec(x + 30°)

Now, we can use the identity cos(x) = 1/sec(x) to rewrite the equation:

1/sec(x) = sec(x + 30°)

1/cos(x) = cos(x + 30°)

cos(x) = cos(x + 30°)

Now, we have two cases to consider:

Case 1: x = x + 30°

This case leads to a contradiction, so it is not a valid solution.

Case 2: x = -x - 30°

This case leads to the equation 2x = -30°, which gives x = -15°.

Therefore, the solution to the equation is x = -15°.