<s>[INST] How to easily solve this trigonometric equation? Given equation:
$$\frac{\sin(x) + \sin(5x) - \sin(3x)}{\cos(x) + \cos(5x) - \cos(3x)} = \tan(3x),$$
what is the easiest way to solve it? I know it can be solved by expanding each $\sin(nx)$ and $\cos(nx)$ terms, but is there an easier way?
 [/INST] $$\frac { 2\sin { \left( \frac { x+5x }{ 2 }  \right) \cos { \left( \frac { x-5x }{ 2 }  \right) -\sin { \left( 3x \right)  }  }  }  }{ 2\cos { \left( \frac { x+5x }{ 2 }  \right) \cos { \left( \frac { x-5x }{ 2 }  \right)  } -\cos { \left( 3x \right)  }  }  } =\frac { \sin { \left( 3x \right) \left( 2\cos { 2x-1 }  \right)  }  }{ \cos { \left( 3x \right) \left( 2\cos { 2x-1 }  \right)  }  } =\tan  (3x)$$
</s>