Prove the identity and give its domain.

4sinxcosx+sin4x=8sinxcos^3x

Use the identities

sin 2y = 2 sin y cosy,
cis 2y = cos^2 y - sin^2 y. and
1 - sin^2 y = cos^2 y
as follows:
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4sinxcosx + sin4x = 8sinxcos^3x
2sin2x + sin4x = (8sinxcosx)cos^2x
2sin2x + 2sin2x cos2x = 4 sin2x cos^2x
Divide both sides by 2sin2x
1 + cos 2x = 2 cos^2 x
1 + cos^2x - sin^2x = 2 cos^2x
Subtract cos^2x from both sides
1 - sin^2 x = cos^2 x
etc.
You may need to prove separately that the identity is valid when sin2x = 0, since we divided by that term.

To prove the given identity, we'll start by simplifying both sides of the equation separately and then equating them.

Given: 4sin(x)cos(x) + sin(4x) = 8sin(x)cos^3(x)

Let's simplify the left side of the equation first:

4sin(x)cos(x) + sin(4x)

Using the double-angle identity for sin(2x) = 2sin(x)cos(x), we can rewrite sin(4x) as 2sin(2x)cos(2x):

4sin(x)cos(x) + 2sin(2x)cos(2x)

Now let's use the double-angle identities for sin(2x) and cos(2x):

4sin(x)cos(x) + 2(2sin(x)cos(x))(cos^2(x) - sin^2(x))

Simplifying further:

4sin(x)cos(x) + 4sin(x)cos(x)(cos^2(x) - sin^2(x))

Factoring out the common term sin(x)cos(x):

4sin(x)cos(x)(1 + cos^2(x) - sin^2(x))

Using the Pythagorean identity sin^2(x) + cos^2(x) = 1:

4sin(x)cos(x)(1 + cos^2(x) - (1 - cos^2(x)))

Simplifying:

4sin(x)cos(x)(1 + cos^2(x) - 1 + cos^2(x))

4sin(x)cos(x)(2cos^2(x))

Now let's simplify the right side of the equation:

8sin(x)cos^3(x)

We can rewrite cos^3(x) as cos(x)cos^2(x):

8sin(x)cos(x)cos^2(x)

Now, let's compare the left and right sides of the equation:

4sin(x)cos(x)(2cos^2(x)) = 8sin(x)cos(x)cos^2(x)

Simplifying further:

8sin(x)cos(x)cos^2(x) = 8sin(x)cos(x)cos^2(x)

Since both sides of the equation are equal, we have proven the given identity. The domain of this identity is all real numbers, as there are no restrictions on the values of x.