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

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

Example 4--Page 2

So what is the second way? You should notice that the only thing that is common on both sides is that they both have cos_2tht.gif (1208 bytes), and this is our key to prove this identity by the second way. Usually we keep things that already are the same on both sides. So we keep cos_2tht.gif (1208 bytes), and try to combine all other terms. And if we want to reduce the differences of operations, which means change addition to multiplication, we need cos_2tht.gif (1208 bytes) to become a common factor, so later we can factor the left side.

Now, how can we get cos_2tht.gif (1208 bytes)? We see a lot of 2’s at left side, like cos_4tht.gif (1198 bytes), cos_6tht.gif (1212 bytes) and there is no argument that we should combine either 6_4201.gif (1355 bytes), or 6_4202.gif (1353 bytes), or 6_4202a.gif (671 bytes), and one of these should work out this problem. We chose 6_4201.gif (1355 bytes), because 6_4201.gif (1355 bytes)6_4203.gif (1462 bytes) which gives cos_2tht.gif (1208 bytes).

Then what do we do to cos_6tht.gif (1212 bytes)? We don't have a cos_6tht.gif (1212 bytes) formula, but we have a Sum formula of 6_4204.gif (1640 bytes) right? Here is what we're going to get:

6_4205.gif (6355 bytes)

We have a little problem here, the last term does not have cos_2tht.gif (1208 bytes)! That's OK, we believe you can figure out that 6_4206.gif (1328 bytes)6_4207.gif (1737 bytes).

6_4301.gif (6865 bytes)


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