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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
strategies for you
Page 1 | Page 2 | Page 3 | Proof(s)

Example 5--Page 3

We know that 6_5301.gif (1709 bytes) must equal to 6_5302.gif (1808 bytes), so that both sides of this identity will be equal. By using a Product to Sum formula we will get it.

6_5303.gif (2891 bytes)

The identity has been proven.

We believe you can understand these proofs very well. So we give a third proof, which is the shortest of these three proofs.

The third proof is very simple. You don't need to do any "borrowing" work. We will prove it from right side to left, because right side has more terms, which usually will allow us to have more stuff to work on.

Very simple, like we said, try to reduce the differences. The left side has no tangent, so first we change tangent functions on the right side.

6_5304.gif (2803 bytes)

The left has no fraction, we want to cancel 6_5305.gif (1209 bytes). Then change the sum of 6_5302.gif (1808 bytes) to product we can get 6_5305.gif (1209 bytes):

6_5306.gif (3636 bytes)

Now, we convert it back to angles 10theta.gif (1057 bytes) and 2theta.gif (991 bytes). Also, to reduce the difference of operations we use the Product to Sum formulas.

6_5307.gif (2197 bytes)

The identity has been proven.

From our three proofs you can see that by using our idea, we can easily know what we should do next. By using our compare and "borrowing" strategies, we can prove identities the way we want. With all these skill, you are ready to see the following examples.


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