Question

Prove the following identity?

I know this is a double angle question, but not sure how to prove it! If you could show me the steps, that would be great!

`sin(3x) = 3sinx - 4sin^3x`

Answer 1

Please see the steps below.

Explanation:

`sin3x=sin(x+2x)`

= `sinxcos2x+cos2xsinx`

= `sinxcos2x+cosxsin2x`

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

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

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

= `sinx-2sin^3x+2sinx-2sin^3x`

= `3sinx-4sin^3x`

Answer 2

See Explanation.

Explanation:

`sin(3x)`
`=sin(2x+x)`
`=sin2xcosx + cos2xsinx`
`=(2 sinxcosx)cosx + (1-2sin^2x)sinx`
`=2sinx(cos^2x)+sinx-2sin^3x`
`=2sinx(1-sin^2x) + sinx - 2sin^3x`
`=2sinx-2sin^3x+sinx-2sin^3x`
`=3sinx - 4sin^3x`

Hope you got it.

Answer 3

Kindly go through a Proof in the Explanation.

Explanation:

`sin3x=ul(sin3x-sinx)+sinx`,

`=2cos((3x+x)/2)sin((3x-x)/2)+sinx`,

`=2cos2xsinx+sinx`,

`=2(1-2sin^2x)sinx+sinx`,

`=(2sinx-4sin^3x)+sinx`.

`rArr sin3x=3sinx-4sin^3x`.