Product-to-Sum / Sum-to-Product Formulas



     The addition and subtraction formulas for sine and cosine can be used to establish identities relating sums and products of sines and cosines. These identities can be used to simplify trigonometric expressions. We begin with the product-to-sum formulas.



Product-to-Sum Formulas

     1.                  (1)

     2.                  (2)

     3.                 (3)

     4.                 (4)



     Let us work through the details of these formulas. To establish (1) and (2), recall that  and .


     Adding these two formulas, notice that the cos Asin B cancel and we have:




     Subtracting the second formula from the first, then the sin Acos B cancel and we have:




     To establish (3) and (4), we shall use the remaining two addition and subtraction formulas:  and .


     If we add these two formulas, the sin Asin B cancel and we have:




     Finally, subtracting the third formula from the fourth, the sin Asin B cancel and we have:





Example 1:


     Evaluate  without using a calculator. Compare this with the value of .




     Using Formula (1) above, we have:




     Formula (4) gives us that




     which is the same value.



Example 2:


     Simplify the expression .




     Using formula (4) and the cofunction formula cos(θ) = sin(90°  θ), we have the following:




     We just saw above that the product-to-sum formulas help to simplify some trigonometric expressions. Other times, though, we may want to go in reverse. That is, we would like to express a sum as a product.


     Let us see how those formulas would go. Before we begin, though, let us make a change of variables. Let  and . Notice that  and .



     With the above substitution, notice that Formula (1) becomes:




     Formula (2) becomes:




     Formula (3) becomes:




     Formula (4) becomes:





     The following table summarizes these results.



Sum-to-Product Formulas

     1.                            (5)

     2.                            (6)

     3.                           (7)

     4.                          (8)



Example 3:


     What is the exact value of ?




     Using formula (7), we have: