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Example 8--Page 5To 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: which has no c. Since , by dividing on both sides, we have: To get the angle , use the Sum to Product formulas. We have:
Since , , from equation (7) we know that Since Hence (6) holds. From equations (5) and (6) we have: 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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