**POINT OF INTEREST 6
DIFFERENCE , DOUBLE, AND HALF ANGLE IDENTITIES
Copyright by Ingrid Stewart, Ph.D. Please Send Questions and Comments to ingrid.stewart@csn.edu.**

**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

, 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,(u v)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

since , , and are points on a unit circle.1Therefore,

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 theSum and Difference Identities forSine.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 Cosinecan be established using .

**Proof of the Double Angle Identities for**

SineTo proof the first identity, let in the

Sum Identity

CosineTo proof the second identity, let in the

Sum Identity

**Proof of the Half Angle Identities for **

SineHere we use the

Double Angle IdentityFinally, let , then

CosineHere we use the

Double Angle IdentityFinally, let , then