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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
ready for the last one?
Page 1 | Page 2 | Page 3 | Page 4 | Page 5 | Proof(s)

Example 8--Page 4

All right, here is the second way. To understand how the second way works, you have to read our Summarize the Formulas section. If you haven't, please go back and take a look at it.

There is one formula in our Summarize the Formulas section:

6_8401.gif (3184 bytes)

where alpha.gif (882 bytes) satisfies 6_8402.gif (1927 bytes), or 6_8403.gif (1159 bytes)(if a = 0). The identity 6_8404.gif (1908 bytes) has 6_8405.gif (1187 bytes), which reminds us that this formula might be able to help us to solve the problem.

By using this formula, we get:

From (1) 6_8406.gif (2265 bytes) (3)         
From (2) 6_8407.gif (2261 bytes) (4)         

When we look at it, we are very happy, because we already got 6_8405.gif (1187 bytes), and at the same time, we have c.

You might want to square the both sides of those equations. Yes, we will do it, but it's not the right time yet, because we know that we want to have 6_8408.gif (1344 bytes), so we will use the Sum to Product formulas. To do that, we add equations (3) and (4).

(3) + (4) 6_8409.gif (3426 bytes)
6_8410.gif (3205 bytes)

Dividing 2 from both sides, we have:

6_8411.gif (5076 bytes)

Now we are almost there. Next we will just square both sides and we have:

6_8412.gif (2878 bytes) (5)         

Again, using our compare strategy, we believe that 6_8412a.gif (1603 bytes) must be equal to 1, so that the identity will be true. We claim that:

6_8413.gif (1791 bytes) (6)         

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