Question

How do you use a half-angle formula to simplify the following expression `(sin(4x))/(1+cos(4x))`?

Answer 1

Use the identity:
`sin(4x)=sin[2(2x)]=2sin(2x)cos(2x)`

and `cos(4x)=cos[2(2x)]= 2cos^2(2x)-1`

`=>cos(4x)=2cos^2(2x)-1 => cos4x+1=cos^2(2x)`

hence, `(sin(4x))/(1+cos(4x)) = (2sin(2x)cos(2x))/(cos^2(2x))=(2sin(2x)cancel(cos(2x)))/(cos(2x)cancel(cos(2x)))`

`=> (sin(4x))/(1+cos(4x))= (2sin(2x))/(cos(2x))=2tan(2x)`