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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 | Page 4 | Page 5 | Proof(s)

Example 8--Page 5

To prove equation (6), compare it with equations (3) and (4). What are the differences? (3) and (4) have a, b and c, but equation (6) has no a, b or c. We subtract equation (4) from equation (3). We get:

6_8501.gif (3120 bytes)

which has no c. Since a2_b2n_0.gif (1540 bytes), by dividing 6_8405.gif (1187 bytes) on both sides, we have:

6_8502.gif (2451 bytes)

To get the angle 6_8503.gif (1377 bytes), use the Sum to Product formulas. We have:

6_8504.gif (2490 bytes) (7)         

Since tht_n_p_2kpi.gif (1673 bytes), 6_8505.gif (1552 bytes), from equation (7) we know that

6_8506.gif (1723 bytes)

Since 6_8507.gif (2785 bytes)

6_8508.gif (1797 bytes)

Hence (6) holds. From equations (5) and (6) we have:

6_8509.gif (1884 bytes)

The identity has been proven.

Do you feel happy right now? If you don't understand this way try to look at it again. We feel the second way is easier to understand, but it requires knowing our formulas, which most people don't. The main idea still is reduce the differences between the two sides of the identity or between the known identities and goal identity, which we need to prove.


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