Example 7

Suppose .
Prove that .

This problem is a conditional identity. A conditional identity can only be proved under certain conditions. For this particular identity, its condition is angles .

Although it has special condition, we still can use our idea to solve it.

To reduce the differences between both sides of the equation, we reduce the difference of operations. We can think it must be relate to the Sum to Product formulas, which have half angles too. So let's combine any two terms at the left. We choose to combine the first and second terms.

Now we have got one thing we need--. We know the right side is multiplication so we can believe that we will get , so that we can factor the left side and make it become multiplication. We can't see how we can get from . Usually at this point, it's where we need the condition(s) to help us get farther. From the condition, we know that , so . By the Derived formulas , we have .