Use the sum-to-product formulas to write the given expression as a product.

A)2 sin 4θ cos θ
B)-2 sin 4θ sin θ
C)-2 cos 4θ cos θ
D)2 cos 4θ sin θ
E)2 cos 4θ cos θ

Question

Use the sum-to-product formulas to write the given expression as a product. ​
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A)​​2 sin 4θ cos θ 
B)​-2 sin 4θ sin θ 
C)-​2 cos 4θ cos θ 
D)​2 cos 4θ sin θ 
E)​​2 cos 4θ cos θ

Quizplus · Accepted Answer

We can use the following sum-to-product formulas:

- sin A + sin B = 2 cos ((A-B)/2) sin ((A+B)/2)
- sin A - sin B = 2 sin ((A-B)/2) cos ((A+B)/2)
- cos A + cos B = 2 cos ((A+B)/2) cos ((A-B)/2)
- cos A - cos B = -2 sin ((A+B)/2) sin ((A-B)/2)

Using (b) and (d) for the first term, and (d) for the second term, we get:

- 2 sin 4θ cos θ = sin (4θ + θ) + sin (4θ - θ) = sin 5θ - sin 3θ
- 2 cos 4θ sin θ = sin (4θ + θ) - sin (4θ - θ) = sin 5θ + sin 3θ

So the expression is equal to:

(sin 5θ - sin 3θ)(sin 5θ + sin 3θ) = -sin^2 3θ + sin^2 5θ = sin^2 2θ - sin^2 3θ = sin 2θ (sin 2θ - sin 3θ)

This simplifies to -2 sin 4θ sin θ, so the answer is (B).

Use the Sum-To-Product Formulas to Write the Given Expression as a Product