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

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.

Hence

Since , we get

or

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

Using the above formulas, we get

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

Same arguments lead to

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

**Example.**Check the identities**Answer.**We will check the first one. the second one is left to the reader as an exercise. We haveHence

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 haveHence

Since , we get

or

**Example.**Verify the identity**Answer.**We haveUsing 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 . ThenUsing 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 thatwhich 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