The user-friendly version of this content is available here.

The following content is copyright (c) 2009-2013 by Goods of the Mind, LLC.

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:

obtain:

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