## ** Double-Angle and Half-Angle Formulas**

Double-Angle and Half-Angle formulas are very useful. For example, rational functions of sine and cosine wil be very hard to integrate without these formulas. They are as follow

**Example.** Check the identities

**Answer.** We will check the first one. the second one is left to the reader as an exercise. We have

Hence

which implies

Many functions involving powers of sine and cosine are hard to integrate. The use of Double-Angle formulas help reduce the degree of difficulty.

**Example.** Write as an expression involving the trigonometric functions with their first power.

**Answer.** We have

Hence

Since , we get

or

**Example.** Verify the identity

**Answer.**We have

Using the Double-Angle formulas we get

Putting stuff together we get

From the Double-Angle formulas, one may generate easily the Half-Angle formulas

In particular, we have

**Example.** Use the Half-Angle formulas to find

**Answer.** Set . Then

Using the above formulas, we get

Since , then is a positive number. Therefore, we have

Same arguments lead to

**Example.** Check the identities

**Answer.** First note that

which falls from the identity . So we need to verify only one identity. For example, let us verify that

using the Half-Angle formulas, we get

which reduces to

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*Mohamed A. Khamsi *

Tue Dec 3 17:39:00 MST 1996

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