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This essential trains for: AMC-10, AMC-12, AIME, Math Kangaroo 9-10.
Use the sum and difference of arcs identities to derive the product-to-sum and sum-to-product identities:
Add these identities:
and introduce the variable substitution:
wherefrom, by solving for x and for y:
Using the substitution we find the identity between the sum of two sine functions and the product of a sine and a cosine. Since the cosine is an even function, the order of the arcs in the subtraction does not matter:
Similarly, we can derive the difference of two sines by subtracting the two identities, instead of adding them:
and use the same variable substitution to obtain:
From the cosine identities:
we obtain, by adding:
Using the same variable substitution as before:
While by subtracting the identities, we obtain:
In the last identity we can absorb the minus sign inside the argument of one of the sine functions (since they are odd functions):
The product-to-sum identities can be derived from the same identities for the sum of arcs. From:
Similarly, we get from the other identities used in the above: