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Example 5Page 3We know that must equal to , so that both sides of this identity will be equal. By using a Product to Sum formula we will get it. The identity has been proven. We believe you can understand these proofs very well. So we give a third proof, which is the shortest of these three proofs. The third proof is very simple. You don't need to do any "borrowing" work. We will prove it from right side to left, because right side has more terms, which usually will allow us to have more stuff to work on. Very simple, like we said, try to reduce the differences. The left side has no tangent, so first we change tangent functions on the right side. The left has no fraction, we want to cancel . Then change the sum of to product we can get : Now, we convert it back to angles and . Also, to reduce the difference of operations we use the Product to Sum formulas. The identity has been proven. From our three proofs you can see that by using our idea, we can easily know what we should do next. By using our compare and "borrowing" strategies, we can prove identities the way we want. With all these skill, you are ready to see the following examples. 

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