College Algebra & Trigonometry, 2018a

450 CHAPTER 10. TRIGONOMETRIC IDENTITIES AND EQUATIONS
We can represent this distance d with the distance formula used to calculate the
distance between two points in the coordinate plane:
The distance between the points (x 1 ,y 1 ) and (x 2 ,y 2 ) is
d = √ (x 2 − x 1 ) 2 +(y 2 − y 1 ) 2
So, in the first diagram the distance d will be:
d = √ (cos α − cos β) 2 +(sinα − sin β) 2
In the second diagram the distance d will be:
d = √ (cos(α − β) − 1) 2 + (sin(α − β) − 0) 2
Since these distances are the same, we can set them equal to each other:
√
(cos α − cos β)2 +(sinα − sin β) 2 = √ (cos(α − β) − 1) 2 + (sin(α − β) − 0) 2
We’ll square both sides to clear the radicals:
(cos α − cos β) 2 +(sinα − sin β) 2 = (cos(α − β) − 1) 2 + (sin(α − β) − 0) 2
Next, we’ll rewrite (sin(α − β) − 0) 2 as sin 2 (α − β):
(cos α − cos β) 2 +(sinα − sin β) 2 = (cos(α − β) − 1) 2 +sin 2 (α − β)