How do you verify the identity cosx-1=(cos2x-1)/(2(cosx+1))cosx1=cos2x12(cosx+1)?

1 Answer
Mar 30, 2017

Start with the right side:

(cos2x-1) / (2(cosx+1))cos2x12(cosx+1)

Use the identity cos2x = color(red)(2cos^2x-1)cos2x=2cos2x1

(color(red)(2cos^2x-1)-1) / (2(cosx+1))2cos2x112(cosx+1)

(2cos^2x - 2)/(2(cosx+1))2cos2x22(cosx+1)

Now divide both the numerator and denominator by 2:

(cos^2x - 1) / (cosx + 1)cos2x1cosx+1

Factor the numerator:

((cosx+1)(cosx - 1)) / (cosx + 1)(cosx+1)(cosx1)cosx+1

((cancel(cosx+1))(cosx - 1)) / cancel(cosx+1)

cosx - 1

Which is the left side.

Final Answer