Question

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

Answer 1

`LHS=cos2x+cosx`

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

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

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

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

Answer 2

`"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"`