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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