Prove trigonometric identity

Question

Prove that this given equation is an identity:
Cos 3x = 4cos^3 x -3cosx

Answer 1

To prove that it is an identity, we want to write the left side in another way so that it is only in terms of cosine and matches the right side.

 

cos(3x) = cos(2x + x)

         

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

 

           = (2cos²x - 1)cosx - 2sinxcosxsinx

 

           = 2cos³x - cosx - 2cosxsin²x

 

           = 2cos³x - cosx - 2cosx(1 - cos²x)

 

           = 2cos³x - cosx - 2cosx + 2cos³x

 

           = 4cos³x - 3cosx