Question 167730
*[Tex  \LARGE sin(4x)=cos(x)(4sin(x)-8sin^3(x)) ] ... Start with the given equation



*[Tex  \LARGE sin(4x)=cos(x)4sin(x)(1-2sin(x)) ] ... Factor out *[Tex \LARGE 4sin(x)] (good so far)



*[Tex  \LARGE sin(4x)=4sin(x)cos(x)(1-2sin(x)) ] Rearrange the terms.



*[Tex  \LARGE sin(4x)=2*2sin(x)cos(x)(1-2sin(x)) ] Factor 4 into 2*2



*[Tex  \LARGE sin(4x)=2sin(2x)(1-2sin(x)) ] Use the identity *[Tex \LARGE 2sin(x)cos(x)=sin(2x)]



*[Tex  \LARGE sin(4x)=2sin(2x)cos(2x) ] Use the identity *[Tex \LARGE 1-2sin(x)=cos(2x)]



Let {{{z=2x}}}



*[Tex  \LARGE sin(4x)=2sin(z)cos(z) ] Substitute "z" in for 2x



*[Tex  \LARGE sin(4x)=sin(2z) ] Use the identity *[Tex \LARGE 2sin(z)cos(z)=sin(2z)] (similar to the previous used identity just with a different variable)




*[Tex  \LARGE sin(4x)=sin(2*2x) ] Plug in {{{z=2x}}}



*[Tex  \LARGE sin(4x)=sin(4x) ] Multiply



Since both sides of the equation are equal, this means that we've verified the identity *[Tex  \LARGE sin(4x)=cos(x)(4sin(x)-8sin^3(x)) ]