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Example 3--Page 3
Now, how do we combine the 2nd and 3rd terms together? Well, we all know that . So we want to change tangent to sine and cosine.
Then we combine the last 2 terms.
Please note that the top of the fraction actually is by the Sum and Difference formulas. This might be the trickiest part of this proof, but it is still natural, because Sum formulas can help us to make sum to product--two terms combine to one term.
From some basic definition and the Derived formulas we know that value of the function of an angle plus actually is same as the value of the function of original angle. So we have:
So what we do next? Well, good question. You can go ahead and take 10 seconds to think about what we do next.
OK, let me make this clear, what is our goal? We want to make the top become , bottom become . Genie, I wish... No, no, no, it won't work. We need to think by ourselves. We already got angle at the top, but we still want to change the bottom to angle , which will be close to . So do you want to expand and by using Sum formula? No, don't do that. Since we will not get angle . Let's see, does looks familiar? No? How about this: . Yes, it is one of the Product-to-Sum formulas. , and if we use this formula we will get angle .
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