Question

Write in terms of sine only?

sin(3x) − cos(3x).

Answer 1

`3sinx-4sin^3x+-sqrt(1-sin^2x)(1-4sin^2x)`

Explanation:

Using the identities:
`sin(x+y)= sinx*cosy+cosx+siny`
`cos(x+y)= cosx*cosy-sinx*siny`
`sin2x= 2sinxcosx`
`cos2x= 1-2sin^2x`
`cosx= +-sqrt(1-sin^2x)`

Start:
`sin(3x) − cos(3x)=`

`sin(2x+x)-cos(2x+x)=`

`sin2x*cosx+cos2x*sinx-(cos2x*cosx-sin2x*sinx)=`

`2sinxcosx*cosx+(1-2sin^2x)*sinx-((1-2sin^2x)*cosx-2sinxcosx*sinx)=`

`2sinxcos^2x+sinx-2sin^3x-cosx+2sin^2xcosx+2sin^2xcosx=`

`2sinx(1-sin^2x)+sinx-2sin^3x-cosx+4sin^2xcosx=`

`2sinx-2sin^3x+sinx-2sin^3x-cosx+4sin^2xcosx=`

`3sinx-4sin^3x-cosx(1-4sin^2x)=`

`3sinx-4sin^3x+-sqrt(1-sin^2x)(1-4sin^2x)`