Double-Angle and Half-Angle Formulas 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