Y=sin x sin 4x - cos x cos 4x

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To simplify the expression Y = sin(x)sin(4x) - cos(x)cos(4x), we can use the angle difference formula for cosine:

cos(a-b) = cos(a)cos(b) + sin(a)sin(b)

Applying this formula, we have:

Y = sin(x)sin(4x) - cos(x)cos(4x)
Y = sin(x)sin(4x) - cos(x)cos(4x)

Rewriting sin(4x) and cos(4x) using the angle sum formula:

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

Now, substitute these expressions into the equation:

Y = sin(x) 2sin(2x)cos(2x) - cos(x) (2cos^2(2x) - 1)

Expand the terms:

Y = 2sin(x)sin(2x)cos(2x) - 2cos(x)cos^2(2x) + cos(x)

Using trigonometric identities:

sin(2x) = 2sin(x)cos(x)
cos^2(2x) = 1 - 2sin^2(2x)

Substitute these identities into the equation:

Y = 2sin(x) * 2sin(x)cos(x)cos(2x) - 2cos(x)(1 - 2sin^2(2x)) + cos(x)

Simplify the terms further:

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

Y = 4sin^2(x)cos(x)cos(2x) + 4sin^2(2x)cos(x) - 3cos(x)

Therefore, the simplified expression for Y = sin(x)sin(4x) - cos(x)cos(4x) is:

4sin^2(x)cos(x)cos(2x) + 4sin^2(2x)cos(x) - 3cos(x)