Product-Sum and Sum-Product Identities

These identities are used to convert product forms to sum forms,
a change that is sometimes a good idea when working certain problems,
for example, in the calculus.


                    Product-Sum Identities

These are easy to obtain.
Simply add or subtract the appropriate sum and difference identities.
For example, add the following two identities

     
     
to get

     
Divide both sides of the equation by to get the identity for the sine function.
Including this identity, similar identities are:


          
          
          
          

These identities express product involving sines or cosines as sums involving sines or cosines.

     See Examples 1 – 2, pages 552 – 553, of the textbook.


                    Sum-Product Identities

The above product-sum identities easily an be rewritten
to form the sum-product identities.
Here is an example of a typical derivation of one of the sum-product identities.

Start with the product-sum identity

     

We want to rewrite the right side of the equation
so that we have the sum of the sines of single quantities.

So, let
     
     

Using either the method of substitution or the method of elimination, solve this system of linear equation
for and .

We get
          
          

Substituting these expressions into the product-sum-sum identity above, we get



Other sum-product identities are

     
     
     

          See Examples 3 – 4, page 554, of the textbook.


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