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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 3

Comparing the two sides of equation (6), the left side has angle tht_p_phi_2.gif (1119 bytes) the right

side has no angle tht_p_phi_2.gif (1119 bytes). So we want to cancel the angle tht_p_phi_2.gif (1119 bytes).

In equation (6), expand the square. We get:

6_8301.gif (4286 bytes) (10)         

Comparing this with the right side of equation (6), the right side has no term of form 2ab, so we want to cancel the term of 6_8302.gif (2313 bytes).

Comparing equations (6) and (9) reminds us of the algebraic formulas 6_8303.gif (1825 bytes)6_8304.gif (1593 bytes), which cancels the middle term 2xy in the perfect squares.

We square equation (9) and add it with equation (10). Because it equals to 0, so we are adding nothing. After combining like terms, we will cancel the terms 6_8305.gif (2188 bytes).

6_8306.gif (11004 bytes)

The formula sin_cos1.gif (1861 bytes) can help us to cancel the angle tht_p_phi_2.gif (1119 bytes), we have:

6_8307.gif (1394 bytes), that is, (6) holds.

Now let's go back to equation (5), we have:

6_8308.gif (2206 bytes)

Since a2_b2n_0.gif (1540 bytes),

6_8310.gif (1901 bytes)

The identity has been proven.

So how do you feel now? If necessary, feel free to go back and look at it again.

Actually by using our idea—reduce the differences, we can easily explain why we took each step, even on hard problems. And believe it or not, we have another way to prove this identity. The second way is neither harder nor easier. It will just add more powerful skills to you.


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