College Algebra & Trigonometry, 2018a

10.2. DOUBLE-ANGLE IDENTITIES 453
If we divide by 2 on both sides, we’ll have:
1 − cos α cos β − sin α sin β =1− cos(α − β)
then subtract 1
− cos α cos β − sin α sin β = − cos(α − β)
and multiply through by −1
cos α cos β +sinα sin β =cos(α − β)
So, cos(α − β) =cosα cos β +sinα sin β.
This will help us to generate the double-angle formulas, but to do this, we don’t
want cos(α − β), we want cos(α + β) (you’ll see why in a minute).
So, to change this around, we’ll use identities for negative angles. Recall that
in the fourth quadrant the sine function is negative and the cosine function is
positive. For this reason, sin(−θ) =− sin(θ) and cos(−θ) = cos(θ).
Now we can say that cos(2θ) = cos(θ + θ) = cos(θ − (−θ)). Going back to our
identity for cos(α − β), we can say that:
cos(θ − (−θ)) = cos θ cos(−θ)+sinθ sin(−θ)
cos(θ − (−θ)) = cos θ cos θ +sinθ(− sin θ)
cos(θ − (−θ)) = cos θ cos θ − sin θ sin θ
cos(θ − (−θ)) = cos 2 θ − sin 2 θ
cos(θ + θ) =cos 2 θ − sin 2 θ
cos(2θ) =cos 2 θ − sin 2 θ
This is the double-angle identity for the cosine: cos(2θ) =cos 2 θ − sin 2 θ.