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
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#

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.

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#.