Question

Express this product as a sum or difference. cos(9x)cos(2x)

Answer 1

The product
`\( \cos(9x)\cos(2x) \)` can be expressed as the sum \
`( (1)/(2)[\cos(11x) + \cos(7x)] \).`

Step-by-step explanation:

To express the product
`\( \cos(9x)\cos(2x) \)` as a sum or difference, we can use the trigonometric identity
`\( \cos(A)\cos(B) = (1)/(2)[\cos(A+B) + \cos(A-B)] \)` . In this case, A = 9x and B = 2x . Applying the identity, we get
`\( (1)/(2)[\cos(11x) + \cos(7x)] \).`

The identity is derived from the angle addition formula for cosine, which states that
`\( \cos(A + B) = \cos(A)\cos(B) - \sin(A)\sin(B) \)` . By letting A = 9x and B = 2x , we obtain
`cos(9x + 2x) = cos(11x) = cos(9x)cos(2x) - sin(9x)\sin(2x) \)` . Simplifying and rearranging terms, we arrive at the expression
`\( (1)/(2)[\cos(11x) + \cos(7x)] \).`

This trigonometric simplification is often useful in calculus and physics when dealing with products of trigonometric functions. Expressing such products as sums or differences helps simplify calculations and makes it easier to analyze and understand the behavior of the functions involved.

Answer 2

The product cos(9x)cos(2x) is expressed as a sum of two cosine functions using the cosine product-to-sum identity, resulting in ½[cos(11x) + cos(7x)].

Step-by-step explanation:

The product of cos(9x)cos(2x) can be expressed as a sum or difference using the cosine product-to-sum identities. These identities are derived from trigonometric formulas and help to simplify the product of two cosine functions into a sum or difference of two cosine functions. The relevant identity for this problem is:

cos a cos b = ½[cos(a + b) + cos(a - b)]

Applying this identity, we have:

cos(9x)cos(2x) = ½[cos(9x + 2x) + cos(9x - 2x)]

cos(9x)cos(2x) = ½[cos(11x) + cos(7x)]

This expression simplifies the product of two cosines into the sum of two cosine functions, which is easier to work with in many mathematical contexts.