Tanxsin2x=0 factor

To solve the equation tan(x)sin(2x) = 0, we need to find the values of x for which either tan(x) or sin(2x) is equal to zero.

First, let's consider the case when tan(x) = 0.
This occurs when x is equal to multiples of π, so x = nπ, where n is an integer.

Now, let's consider the case when sin(2x) = 0.
Using the double angle identity for sine, sin(2x) = 2sin(x)cos(x).
Setting 2sin(x)cos(x) = 0, we have two possibilities:
1. sin(x) = 0, which occurs when x is equal to multiples of π, so x = nπ, where n is an integer.
2. cos(x) = 0, which occurs when x = (2n + 1)(π/2), where n is an integer.

Combining all the solutions, we have:
x = nπ, (2n + 1)(π/2), where n is an integer.