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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:

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Add these identities:

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and introduce the variable substitution:

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wherefrom, by solving for x and for y:

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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:

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Similarly, we can derive the difference of two sines by subtracting the two identities, instead of adding them:

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and use the same variable substitution to obtain:

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From the cosine identities:

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we obtain, by adding:

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Using the same variable substitution as before:

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While by subtracting the identities, we obtain:

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In the last identity we can absorb the minus sign inside the argument of one of the sine functions (since they are odd functions):

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The product-to-sum identities can be derived from the same identities for the sum of arcs. From:

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obtain:

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Similarly, we get from the other identities used in the above:

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