Question

How do you simplify the expression 6sin(x)cos³x - 6sin³x cos(x)?

a) 6sin(4x)
b) 6sin(2x)
c) 3sin(2x)
d) 3cos(2x)

Answer 1

The expression 6sin(x)cos³(x) - 6sin³(x)cos(x) simplifies to 3/2 sin(4x) using trigonometric identities, but this result does not match the provided choices, suggesting an error in the question or choices.

Step-by-step explanation:

To simplify the expression 6sin(x)cos³(x) - 6sin³(x)cos(x), we can factor out the common terms and apply trigonometric identities.

First, factor out the common term 6sin(x)cos(x):

6sin(x)cos(x)(cos²(x) - sin²(x))

Then use the double-angle formula for cosine, which is cos(2x) = cos²(x) - sin²(x):

6sin(x)cos(x)cos(2x)

Now apply the double-angle formula for sine, which is sin(2x) = 2sin(x)cos(x):

6sin(x)cos(x)cos(2x) = 6 * 1/2 * sin(2x) * cos(2x) = 3sin(2x)cos(2x)

Finally, apply the double-angle formula again, which is sin(2x) = 2sin(x)cos(x), we get:

3 * 1/2 * sin(4x) = 3/2 sin(4x)

However, this result does not match any of the options provided in the question. Hence, it's likely that there is a misunderstanding of the question's requirements, or there is an error in the provided choices.