Solve the following for all values of $x$ in the domain $(0, 2π)$. Use exact values.

1. $\text{sin}(2x) =\text{sin}(x)$

   Hint (a):

   $\text{sin}(2x) = 2\text{sin}(x)\text{cos}(x)$. Substitute and solve for $x$.

   Answer (a):

   There will be four solutions.

2. $\operatorname { sin } ( x + \pi ) + \operatorname { cos } ( x - \frac { \pi } { 2 } ) = 1$

   Hint (b):

   $\text{sin}(x+\pi )+\text{cos}\left(x-\frac{\pi}{2}\right)=-\text{sin}(x)+\text{cos}(x)\text{cos}\left(\frac{\pi}{2}\right)+\text{sin}(x)\text{sin}\left ( \frac{\pi}{2} \right )$

   Answer (b):

   There is no solution to this problem.

3. $\operatorname { cot } x - \operatorname { tan } x = 2 \sqrt { 3 }$

   Hint (c):

   $\text{cot}(x)=\frac{\text{cos}(x)}{\text{sin}(x)}\ \ \ \ \ \text{tan}(x)=\frac{\text{sin}(x)}{\text{cos}(x)}$

   Answer (c):

   There will be four solutions.