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Example 1
Example 2
Example 3
Example 4
Example 5
Example 6
Example 7
Example 8
General ideas

   Final thoughts

General ideas

In general, the strategies to prove an identity are as follows.

  1. Find the differences in angles, operations and functions of the two sides of the equation.

  2. Decide which difference you want to reduce first. You might find you are reducing more than one difference at a time while you work. That's fine.

  3. Think about what kind of formulas that can be used to reduce the differences, or what kind of strategies you can use. Choose appropriate formulas to reduce the differences. Remember that you can use formulas from either side. As long as you are reducing the differences, you are on the right way. If you find you are not reducing the differences, you need be careful that you know what you are going to get.

  4. Comparing the result you get at every step with the other side of the identity. Find out what the differences are, and continue the verification.

  5. For conditional identities, you should use all the conditions, and compare the two sides of the goal identity, which you need to prove. You should also compare the goal identity with the known conditional identity if there are any. If you can't go farther when you are proving the identity, it could be the time for you to use the conditions.

  6. When all the differences of the two sides of the equation are canceled, then the verification is done.

While you work more and more on problems, using our idea, you will find more and more strategies. You might even solve some problems you can't ever imagine, either in mathematics or in the real world. We honestly wish you to have fun while doing them.

Now, you have finished our Examples section, do you understand our ideas? Do you know all the formulas? Do you understand our examples and know why we took each step? We hope that you very well understand how to discover some clues in a problem and how to choose some formulas to solve the problem by using the clues. If so, we are very happy to congratulate you that you have finished our Learning sections.


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