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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 | Proof(s)

Example 1

6_1101.gif (3609 bytes)

We see that the functions of this identity are different, as well as the operations and angles. Sometimes we should reduce the differences of angles first, but this is not always. The big difference between both sides of this identity is difference of operations. The left side has addition and division, the right side has multiplication. What we should do is combine the left side to one term as multiplication, which will reduce the difference of operations. After we combine the left side we have:

6_1102.gif (3699 bytes)

Very natural, we need to continue simplifying it.

6_1103.gif (5997 bytes)

As we know, the angle of the right side is 2theta.gif (991 bytes). So the next step is to change angles on the left side to 2theta.gif (991 bytes). By using Double Angle formulas, 6_1104.gif (2067 bytes) and 6_1105.gif (2173 bytes). We get:

6_1106.gif (2619 bytes)

The identity has been proven.

At the last step, without thinking we will change 6_1107.gif (1359 bytes) to tan_2tht.gif (1236 bytes), but actually, we are reducing the differences of functions.

We know that some people might think without using our idea, they still can prove this identity. Yes, it's true, but most of them can't explain how they solve them. In this Examples section, when you see some hard problem, you will know how our idea helps.


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