POINT OF INTEREST 6
DIFFERENCE , DOUBLE, AND HALF ANGLE IDENTITIES Proof of the Difference Identity for Cosine We will use the following picture for this proof: Since the angles between the segment OA and OC and the segment OB and OD both measure (u v), the arcs AC and BD must have the same length. This implies that the line segments connecting points A and C and points B and D are also equal in length.  That is, Raising both sides to the second power and combining like terms, we get Notice that the parentheses contain circles. The value of the three quantities within the parentheses is 1 since , , and are points on a unit circle.

Therefore, Finally, letting     we get , which is usually presented as Proof of the Sum and Difference Identities for Sine and Given the Difference Identity for Cosine, it is easy to show that . We will use this fact to show a proof of the Sum and Difference Identities for Sine.

If we let , then we can prove a Sum Identity for Sine! If we let , then we can prove a Difference Identity. Incidentally, the Sum Identity for Cosine can be established using .

Proof of the Double Angle Identities for

Sine

To proof the first identity, let in the Sum Identity  Cosine

To proof the second identity, let in the Sum Identity  Proof of the Half Angle Identities for

Sine

Here we use the Double Angle Identity  Finally, let , then Cosine

Here we use the Double Angle Identity  Finally, let , then 