1985 AHSME Problems/Problem 16
Problem
If and , then the value of is
Solution
Solution 1
First, let's leave everything in variables and see if we can simplify .
We can write everything in terms of sine and cosine to get .
We can multiply out the numerator to get .
It may seem at first that we've made everything more complicated, however, we can recognize the numerator from the angle sum formulas:
Therefore, our fraction is equal to .
We can also use the product-to-sum formula
to simplify the denominator:
.
But now we seem stuck. However, we can note that since , we have , so we get
Note that we only used the fact that , so we have in fact not just shown that for and , but for all such that , for integer .
Solution 2
We can see that . We also know that . First, let us expand .
We get .
Now, let us look at .
By the sum formula, we know that
Then, since , we can see that
Then
Thus, the sum become and the answer is
See Also
1985 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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