Be sure that you've familiarized yourself with the second set of ...

Question

Be sure that you've familiarized yourself with the second set of formulas presented in this section by working C5–C8 in the Concept and Vocabulary Check. In Exercises 9–22, express each sum or difference as a product. If possible, find this product's exact value.

3x x
cos -------- + cos --------
2 2

Accepted Answer

Hello, everyone. We are gonna write the given trigonometric sum as a product. The sum we are given is the cosine of seven X divided by two plus the cosine of five X divided by two. We are given four answer choices. First thinking about the sum to a product formula. When we are working with cosine, I recall that that is the cosine of A plus. The cosine of B equals two, multiplied by the cosine of A plus B divided by two, multiplied by the cosine of A minus B divided by two. So matching this up with what we're given in our example, A equals seven X divided by two and B equals five X divided by two. So we're gonna substitute those in to our formula. So we will have two multiplied by the cosine of seven X divided by two plus five X divided by two, all of which is divided by two and then multiplied by the cosine of seven X divided by two minus five X divided by two, all of which is divided by two. I want to simplify the numerators here and then I'm gonna simplify the fractions. So this will again, equal two, multiplied by the cosine of be X divided by two, all of which is divided by two. And that will be multiplied by the cosine of two X divided by two, all of which is divided by two. Before I go any further, I notice the fraction that is in my numerators, those both of those fractions can be reduced. So I'm gonna reduce that before I do anything else. So I will have two multiplied by the cosine of six X divided by two, multiplied by the cosine of X divided by two. This looks a lot better already. Now, let's see if there's anything else I can simplify before we say we're done. Um That first cosine we have six X divided by two, we can reduce that fraction. So we will have two multiplied by the cosine of this will be three X and then that will be multiplied by the cosine of X divided by two. This should be our product as simplified as it gets. And so we have two multiplied by the cosine of three X multiplied by the cosine of X divided by two. And this matches answer choice B have a nice day.

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