Question

Solve for `x`? `2sinxcosx = 2sinx`

Answer 1

`x=0` is one of infinitely many solutions. The full solution set is
`{..., –2pi, –pi, 0, pi, 2pi, ...}`

Explanation:

`2sinxcosx=2sinx`

Divide out the 2:

`sinxcosx=sinx`

Move the `sinx` to the left side:

`sinxcosx-sinx=0`

Factor out `sinx`:

`sinx(cosx-1)=0`

For this to be true, we can have `sinx=0` or `cosx-1=0`.

`sinx=0`
`" "=>" "x=npi," "n in ZZ`
`cosx-1=0" "=>" "cosx=1`
`" "=>" "x = 2n*pi," "n in ZZ`

The union of these two solutions is `x = npi," "n in ZZ`.

Here is a graph of `2sinxcosx - 2sinx`. All values of `x` that cross the `y`-axis are our solutions.
graph{2sinx(cosx-1) [-10, 10, -5, 5]}