**SUM AND DIFFERENCE IDENTITIES**

**DOUBLE ANGLE AND HALF ANGLE IDENTITIES**

Example 1:

Show that can be changed to

sin.Let's use the

Difference Identity for Cosine.Here,

uandv.

Your calculator must be in radian mode!We showed that equals

.sin

Example 2:

Show that can be changed to .

Let's use the

Sum Identity for Sine.Here,

uandv.We showed that equals.

Example 3:

Simplify .

Let's use the

SumandDifference Identities of Cosineand .

Here,

uandv.x

We showed that can be simplified to .

Example 4:

Simplify using the

Double Angle Identity.Let's factor out a

to get .3We showed that can be written as

3 sin2x.

Example 5:

Simplify using the

Double Angle Identity.Let's factor out a

to get4We showed that can be written as

.4 cos2x

Example 6:

Simplify using the

Double Angle Identity.Here,

u.xLet's multiply the parentheses using FOIL to get

We showed that can be written as

.cos 2x

Example 7:

Simplify using the

Half Angle Identity.We can see that

, thereforeu 6x.3xUsing the pattern of the

Half Angle Identity, we can write as.sin 3x

Example 8:

Simplify using the

Half Angle Identity.We can see that

, thereforeu 4x.2xUsing the pattern of the

Half Angle Identity, we can write as.cos 2x

Example 9:

Rewrite in terms of a

Double Angle Identityand aHalf Angle Identity:Using :

We can see that

, therefore2u 6x.u 3xUsing the pattern of the

Double Angle Identity, we can write as.2 sin 3x cos 3xUsing :

We can see that

, therefore6x.u 12xUsing the pattern of the

Half Angle Identity, we can write as .