Write 3theta - sin 4theta as a product.

I assume you meant (using x)

sin3x - sin4x

Use the sum-to-product formula:

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

To write 3θ - sin(4θ) as a product, we can use the trigonometric identity:

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

In this case, let A = 4θ and B = 0, then we have:

3θ - sin(4θ) = 3θ - 2sin(2θ)cos(2θ)

Now, we can use another trigonometric identity:

sin(2θ) = 2sin(θ)cos(θ)

Substituting this into the equation, we get:

3θ - 2sin(2θ)cos(2θ) = 3θ - 2(2sin(θ)cos(θ))cos(2θ)

Now, let's combine the terms and simplify further:

= 3θ - 4sin(θ)cos(θ)cos(2θ)

At this point, we can't simplify further or write it as a single product since we have a product of three terms. However, we have expressed 3θ - sin(4θ) as a sum of two terms.