Derivation of the Addition and Subtraction Formulas

Use the figure below. Notice that \(v\) goes in the positive direction, i.e. the initial leg contains \(b\) and the terminal edge is the \(x\) axis. Thus the point \(b\) has coordinates for \(-v\)

1. Follow the steps below to establish the identity \(\cos(u+v) = \cos u \cos v - \sin u \sin v\).
   1. Find \(c^2\) by using the law of cosines.
   2. Find \(c^2\) by using the distance formula \(c = d(a,b)\).
   3. Combine the two formulas above for \(c^2\) to establish the above identity.
2. Use the even/odd properties for \(\cos\) and \(\sin\) to establish the identity \(\cos(u-v) = \cos u \cos v + \sin u \sin v\) (Hint: \(\cos(u-v) = \cos(u + (-v))\)).
3. Use the fact that \(\sin(u+v) = \cos(\pi/2 - (u+v))\) to establish the identity \(\sin(u+v) = \sin u \cos v + \cos u \sin v\).
4. Use the fact that \(\sin(u-v) = \sin(u + (-v))\) to establish the identity \(\sin(u-v) = \sin u \cos v - \cos u \sin v\).
5. Using the addition and subtraction formulas, find \(\sin(15^\circ)\).
6. Using the addition and subtraction formulas, find \(\cos(\pi/12)\).