How do you prove that cos2x+cosx=(cosx+1)(2cosx-1)?

2 Answers
May 26, 2018

LHS=cos2x+cosx

=2cos^2x-1+cosx

=2cos^2x+2cosx-cosx-1

=2cosx(cosx+1)-1(cosx+1)

=(cosx+1)(2cosx-1)=RHS

May 26, 2018

"see explanation"

Explanation:

"using the "color(blue)"trigonometric identity"

•color(white)(x)cos2x=2cos^2x-1

"consider the left side"

2cos^2x-1+cosx

=2cos^2x+cosx-1

"this is a quadratic in cos"

"the factors of the product "2xx-1=-2

"which sum to + 1 are + 2 and - 1"

"split the middle term using these factors"

2cos^2x+2cosx-cosx-1larrcolor(blue)"factor by grouping"

=2cosx(cosx+1)-1(cosx+1)

=(cosx+1)(2cosx-1)

="right side"rArr"verified"