cos(3x)= cos^3x-3 sin^2x cos x

Question

verify identity step by step

Answer 1

Work out the left side.

 

cos(3x) =

cos(2x + x)

 

Use addition-angle identity for cosine.

 

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

 

(1 - 2sin²x)cosx - 2sinxcosxsinx =

 

cosx - 2sin²xcosx - 2sin²xcosx =

 

cosx - 4sin²xcosx

 

 

Does not verify.

Answer 2

Hi Devin,

 

cos3x = cos(2x+x)

cos(2x+x) = cos2xcosx - sin2xsinx

cos2x = cos^2x - sin^2x  and  sin2x = 2sinxcosx

cos(2x+x) = (cos^2x - sin^2x)cosx - 2sinxcosxsinx

= cos^3x - sin^2xcosx - 2sin^2cosx

= cos^3x - 3sin^2xcosx