Question

Prove that ??`(Sinx+Sin2x+Sin3x)/(cosx+cos2x+cos3x) = tan2x`

Answer 1

`LHS=(sinx+sin2x+sin3x)/(cosx+cos2x+cos3x)`

`=(2sin((3x+x)/2)*cos((3x-x)/2)+sin2x)/(2cos((3x+x)/2)*cos((3x-x)/2)+cos2x`

`=(2sin2x*cosx+sin2x)/(2cos2x*cosx+cos2x)`

`=(sin2xcancel((1+2cosx)))/(cos2xcancel((1+2cosx)))`

`=tan2x=RHS`