College Algebra & Trigonometry, 2018a

452 CHAPTER 10. TRIGONOMETRIC IDENTITIES AND EQUATIONS
Now that we’ve simplified the left-hand side, we’ll simplify the right-hand side.
First we’ll square the expression (cos(α − β) − 1) 2 :
(cos(α − β) − 1) 2 =cos 2 (α − β) − 2cos(α − β)+1
So, the right-hand side is now:
cos 2 (α − β) − 2cos(α − β)+1+sin 2 (α − β)
If we rearrange this expression, we’ll again have a nice Pythagorean Identity:
sin 2 (α − β)+cos 2 (α − β) − 2cos(α − β)+1
1 − 2cos(α − β)+1
2 − 2cos(α − β)
2(1 − cos(α − β))
So the left-hand side was equal to:
2(1 − cos α cos β − sin α sin β)
And the right-hand side was equal to:
2(1 − cos(α − β))
So, our original statement in simplified form is:
2(1 − cos α cos β − sin α sin β) =2(1− cos(α − β))