Add or subtract, instead of multiplying, trig functions
8.28 List the product-to-sum formulas that correspond with each of the following four products: cos α cos β, sin α sin β, cos α sin β, and sin α cos β.
The product-to-sum formulas are not typically used to simplify a product, as the equivalent sums are usually lengthier. They are specifically designed to rewrite a product of sine and/or cosine expressions as a sum of sine or cosine expressions.
Notice that a product of two matching trigonometric functions (cos α cos β or sin α sin β) translates into a sum containing cosine functions. Alternately, products that include different trigonometric functions (cos α sin β and sin α cos β) translate into a sum containing sine functions.
8.29 Verify the product-to-sum formula for sin α sin β.
According to Problem 8.28, 
. Apply the sum and difference formulas for cosine to expand the right side of the equation and verify the statement.
If you need to review these sum and difference formulas, flip back to Problem 7.37.
8.30 Write the following product as a sum or difference: cos x sin 2x.
Apply the cos α sin β product-to-sum formula presented in Problem 8.28. Note that α = x and β = 2x.
According to a negative identity presented in Problem 7.14, sin (–x) = –sin x.
8.31 Verify the statement: 
.
Notice that cos2 4x = (cos 4x)(cos 4x). Apply the cos α cos β product-to-sum formula to expand the left side of the equation, such that α = 4x and β = 4x.
Multiply both sides of the equation by 2 in order to eliminate the fractions, and recall that cos 0 = 1 (according to the unit circle).
Both sides of the equation are equal—the order in which you add does not affect the sum according to the commutative property of addition—so the original statement is verified.