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[Index]

Introduction
Essence
Memorizing
Understanding
Summarizing
Examples
Exercises
Final thoughts

## 8. Final thoughts

Balance is a kind of beauty, especially when you find balance between huge differences.

The procedure of proving trigonometric equality is a procedure of building balance, so the procedure we have been thought is a procedure to build beauty.

Our idea is very simple and natural in previous discussions. That is getting a clear understanding of the differences of the both sides in an identity, finding the clues and getting the inspiration from the differences, reducing the differences and finishing the procedure of the proof. Such thinking methods not only can be applied to trigonometric identities but also to more general mathematics areas. We don't want this web site to be too long, so we just stop here.

Maybe saying our web site is a philosophical reading, or a reading material about learning methods and thinking methods better than say it is about mathematical reading material. Our philosophical ideas are:

1. There are thousands of trigonometric identities. Formally they are different from each other, essentially they have common points; that is the two sides of every identity have mainly three kinds of differences: angles, functions and operations. Exactly as people have many kinds of differences: the types of faces, the colors of skins, tall or short, fat or thin, but the structure of the muscles and skeletons in their bodies are the same.

2. The world is complex. Many phenomena are surprises, which look elusive, but we hold that the world can be known. We think that every problem offers some clues, which show us how to solve the problem itself. Our tasks should be discovering these clues. In our text we try to explain how the problems offer some clues, and how we choose suitable formulas to solve the problems by using these clues. We hold that the existence of differences implies the existence of problems. The process of reducing the differences is the process of solving the problem. The keys to solving problems hide in the differences. Analyzing the essences of the differences can help us to get the key, reduce the differences and solve the problem finally.

Mother Nature is beautiful. All things on earth grow up, rely on each other, and restrict each other. Birds eat insects, insects eat trees, trees support birds. If there were no birds, there would be too many insects and trees would be eaten by insects. If there were no trees, the earth would grow sick, how would humans survive? Hence humans need to protect trees and birds. Protecting ecological balance finally protects themselves.

Every day the world economy yields differences. A few people become rich, many people become poor. Depending on how it develops, society may not be stable. So we need a government, we need a president. Let him study how to decrease the rate of unemployment, increase the income of everyone to make society stabilize. All of these things are the same as in our discussion about trigonometric identities!

We thank you for coming to our web site. We hope you have learned something about trigonometric identities from our web site, and are more confident in your ability to deal with them and with other problems. Let beauty and happiness follow you forever. Good luck!

LWR