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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

Suppose that a, b, c, theta.gif (908 bytes), and phi.gif (942 bytes) are real numbers, with the following conditions: a2_b2n_0.gif (1540 bytes), tht_n_p_2kpi.gif (1673 bytes), k is any integer.

6_8101.gif (1956 bytes) (1)
6_8102.gif (1950 bytes) (2)

Prove that 6_8103.gif (1903 bytes).

You might already notice that this is the hardest problem in our entire Learning section. Yes, it's a difficult problem. Especially you might not even know where you should start.

Like the previous one, this is a conditional identity too. We see this problem gives us many more conditions. How do we start? What formulas can help us to prove it? We believe that every problem has some clues. Which tell us how to solve it. Let's see what clues this problem has.

From the left side of the goal identity, we see the angle tht_m_phi_2.gif (1084 bytes), which let us believe it must be relate to the Sum to Product formulas. So let's try to add equations (1) and (2) by using the Sum to Product formulas. We'll get the angle tht_m_phi_2.gif (1084 bytes) and see if we can get anything else.

(1) + (2) 6_8104.gif (3139 bytes) (3)         

From (3), we use the Sum to Product formulas to get:

6_8105.gif (6968 bytes)

Then we divide 2 from both sides and take out the common factor:

6_8106.gif (3075 bytes) (4)         

Let's look at equation (4), because the goal identity has square at both sides, so if we want to get 6_8107.gif (1398 bytes) we need square both sides of equation (4).

6_8108.gif (3381 bytes) (5)         

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