## 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  [Trigonometry]
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