Answer:
First, we will prove the difference formula for cosines. Let’s consider two points on the unit circle. Point \displaystyle PP is at an angle \displaystyle \alphaα from the positive x-axis with coordinates \displaystyle \left(\cos \alpha ,\sin \alpha \right)(cosα,sinα) and point \displaystyle QQ is at an angle of \displaystyle \betaβ from the positive x-axis with coordinates \displaystyle \left(\cos \beta ,\sin \beta \right)(cosβ,sinβ). Note the measure of angle \displaystyle POQPOQ is \displaystyle \alpha -\betaα−β.
Label two more points: \displaystyle AA at an angle of \displaystyle \left(\alpha -\beta \right)(α−β) from the positive x-axis with coordinates \displaystyle \left(\cos \left(\alpha -\beta \right),\sin \left(\alpha -\beta \right)\right)(cos(α−β),sin(α−β)); and point \displaystyle BB with coordinates \displaystyle \left(1,0\right)(1,0). Triangle \displaystyle POQPOQ is a rotation of triangle \displaystyle AOBAOB and thus the distance from \displaystyle PP to \displaystyle QQ is the same as the distance from \displaystyle AA to \displaystyle BB.
Step-by-step explanation:
