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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
good job
Page 1 | Page 2 | Proof(s)

Example 7

Suppose 6_7101.gif (1473 bytes).
Prove that 6_7102.gif (3332 bytes).

This problem is a conditional identity. A conditional identity can only be proved under certain conditions. For this particular identity, its condition is angles 6_7101.gif (1473 bytes).

Although it has special condition, we still can use our idea to solve it.

To reduce the differences between both sides of the equation, we reduce the difference of operations. We can think it must be relate to the Sum to Product formulas, which have half angles too. So let's combine any two terms at the left. We choose to combine the first and second terms.

6_7103.gif (4127 bytes)

What should we do to sin_gam.gif (1097 bytes)? It's hard to figure out. Compare the angles of the two sides, left has angle gamma.gif (919 bytes), and right side has gamma_2.gif (947 bytes). But we know that 6_7104.gif (1186 bytes)6_7105.gif (1550 bytes), so 6_7106.gif (2026 bytes).

6_7107.gif (3054 bytes)

Now we have got one thing we need--6_7108.gif (1174 bytes). We know the right side is multiplication so we can believe that 6_7109.gif (1908 bytes) we will get 6_7108.gif (1174 bytes), so that we can factor the left side and make it become multiplication. We can't see how we can get 6_7108.gif (1174 bytes) from 6_7109.gif (1908 bytes). Usually at this point, it's where we need the condition(s) to help us get farther. From the condition, we know that 6_7101.gif (1473 bytes), so 6_7110.gif (1528 bytes). By the Derived formulas 6_7111.gif (1603 bytes)cos_alph.gif (1101 bytes), we have 6_7112.gif (1417 bytes)6_7113.gif (1603 bytes)6_7108.gif (1174 bytes).

6_7114.gif (4742 bytes)


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