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

3 Answers
Shwetank Mauria
Mar 14, 2018

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

Candy
Mar 14, 2018

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.

Ratnaker Mehta
Mar 14, 2018

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.

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