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