Tìm x, biết: Cot(π/3+x) + cot2x =0

To solve the equation cot(π/3 + x) + cot(2x) = 0, we can use the identity cot(θ) = 1/tan(θ).

First, let's rewrite the equation using the identity:

1/tan(π/3 + x) + 1/tan(2x) = 0

Next, let's find a common denominator for the two terms:

(tan(2x) + tan(π/3 + x))/[tan(π/3 + x) * tan(2x)] = 0

Now, let's simplify the numerator:

tan(2x) + tan(π/3 + x) = 0

Using the identity tan(A + B) = (tan(A) + tan(B))/(1 - tan(A) * tan(B)), we can rewrite the equation:

tan(2x + π/3) = 0

Now, let's find the values of x that satisfy this equation. Since tan(θ) = 0 when θ is an integer multiple of π, we have:

2x + π/3 = nπ, where n is an integer

Solving for x, we get:

2x = nπ - π/3

x = (nπ - π/3)/2

Therefore, the solution to the equation cot(π/3 + x) + cot(2x) = 0 is x = (nπ - π/3)/2, where n is an integer.