6. Examples

So finally here we are. You will learn the most here—the Examples section, because we didn't discuss any real identity in any previous sections. What is it in this section? Well, some very good, or tricky, or you can say hard problems. But don't worry about them, you will be fine. We have very detailed help, and hints. We go through every problem with you here. What you need is patience. Don't fear that you will not understand. You will understand our method. This is a guarantee.

In previous sections, we indicated that there are three differences between the two sides of an identity: functions, operations and angles. The process of reducing the differences is the process of using some basic formulas to transform one side of the identity to the another side. We said every problem has some clues, which will tell us how to solve it. In this section, through eight examples, we will show you how we discover some clues that lead us solve the problem. From the eight examples you will understand what is the correct direction of proving, and which steps you should take.