Final answer:
To solve tan(2x) = 0 within the interval 0 ≤ x ≤ 2π, we look for values where 2x is an integer multiple of π, leading to solutions x = 0 and x = π, as tan(π/2) and tan(3π/2) are undefined.
Step-by-step explanation:
To find all solutions to the equation tan(2x) = 0 on the interval 0 ≤ x ≤ 2π, we need to consider the properties of the tangent function and its periodicity.
The tangent function is zero when its argument is an integer multiple of π.
Therefore, for 2x to be an integer multiple of π, x must be an integer multiple of π/2.
Since we are only looking within the interval from 0 to 2π, the potential solutions where 2x is an integer multiple of π are x = 0, π/2, π, 3π/2, and 2π.
However, tan(π/2) and tan(3π/2) are undefined,
so these values are not solutions to the equation.
Thus, the only solutions within the given interval are x = 0 and x = π.