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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
make it simple
Page 1 | Page 2 | Proof(s)

Example 2

6_2101.gif (2135 bytes)

The difference of angles is sharp. The right side has no theta.gif (908 bytes), so first we want to cancel theta.gif (908 bytes) in left side.

If you read our Memorizing the formulas section, this problem should not be a hard one for you because we included three Triple Angle formulas there. Two of these we will use here. Namely, 6_2102.gif (1323 bytes)6_2103.gif (1816 bytes) and 6_2105.gif (1761 bytes).

We see we can factor sin_tht.gif (342 bytes) from sin_3tht.gif (1224 bytes), and cos_tht.gif (344 bytes) from cos_3tht.gif (1208 bytes). Thus we can simplify 6_2106.gif (1297 bytes) and 6_2107.gif (1297 bytes). By using the Triple Angle formulas we get:

6_2108.gif (3062 bytes)

Being very straight forward, we want to factor out sin_tht.gif (342 bytes) and cos_tht.gif (344 bytes), at the top of the two fractions respectively, and cancel it with the bottom.

6_2109.gif (4896 bytes)

Then combine the like terms, we have

6_2110.gif (2097 bytes)

We know sin_cos1.gif (1861 bytes) can help us to cancel theta.gif (908 bytes), so we factor out -4, we get,

6_2111.gif (3142 bytes)

The identity has been proven.

It's not hard right? Without knowing our idea you can do it too, but can you give another proof?


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