Question

Use the product-to-sum identities to rewrite the following expression as a sum or difference:

3sin(x)sin(3x)
a) cos(2x)−cos(4x)
b) cos(4x)−cos(2x)
c) cos(2x)+cos(4x)
d) cos(4x)+cos(2x)

Answer 1

The expression 3sin(x)sin(3x) can be rewritten using the product-to-sum identities as (a) cos(2x)−cos(4x). This is achieved by applying the formula sin(a)sin(b) = 1/2[cos(a-b) - cos(a+b)].

Step-by-step explanation:

To rewrite the expression 3sin(x)sin(3x) using the product-to-sum identities, we can use the following formula, which is a variation of the product-to-sum identity:

sin(a)sin(b) = 1/2[cos(a-b) - cos(a+b)]

Applying this formula to our expression:

3sin(x)sin(3x) = 3/2[cos(x-3x) - cos(x+3x)]

Simplify the angles inside the cosines:

3sin(x)sin(3x) = 3/2[cos(-2x) - cos(4x)]

Since cos(-θ) = cos(θ), we have:

3sin(x)sin(3x) = 3/2[cos(2x) - cos(4x)]

Now we divide the entire equation by 3 to match the format of the options:

sin(x)sin(3x) = 1/2[cos(2x) - cos(4x)]

Therefore, the correct answer is option (a) cos(2x)−cos(4x), which is rewritten as a difference of cosines.