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   Introduction Introduction
   What's the essence? Essence
   Memorizing Memorizing
   Understanding Understanding
   Summarizing Summarizing
   Examples Examples

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

   Exercises Exercises
   Final thoughts Final thoughts
Page 1 | Page 2 | Page 3 | Page 4 | Proof(s)

Example 3--Page 4

What we will do is multiply by 4 at the top and bottom, so it will cancel those 1_o_2.gif (960 bytes)’s.

6_3401.gif (2370 bytes)

Next we want to combine these two terms to get one term (product) and angle 3theta.gif (1006 bytes). If we change tan_tht.gif (359 bytes) to 6_3402.gif (1244 bytes) and combine the first and second terms, we have:

6_3403.gif (6214 bytes)

Please note that the Product-to-Sum formulas 6_3404.gif (1686 bytes)6_3405.gif (1573 bytes)6_3406.gif (1594 bytes)6_3407.gif (1580 bytes), and 6_3408.gif (1669 bytes)6_3409.gif (1653 bytes), and 6_3410.gif (1643 bytes)6_3411.gif (1691 bytes) all help us to get angle 3theta.gif (1006 bytes). Now we have:

6_3412.gif (7039 bytes)

The identity has been proven.

From the second proof, we can see that we used a lot of proof strategies. They are all based on our idea--reduce the 3 differences. Our idea is simple but powerful.


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Copyright 1998 LWR, ThinkQuest team 17119. All rights reserved.