To simplify this expression, we will use trigonometric identities to rewrite it in terms of sine and cosine functions.

Given expression: Y = sin^2(3x) - 6cos^2(x) + 2

First, recall the Pythagorean identity:
sin^2(x) + cos^2(x) = 1

From this, we can rewrite sin^2(x) in terms of cos^2(x):
sin^2(x) = 1 - cos^2(x)

Now we rewrite the given expression:
Y = (1 - cos^2(3x)) - 6cos^2(x) + 2
Expand and simplify:
Y = 1 - cos^2(3x) - 6cos^2(x) + 2

Next, we use the double angle identity for cosine:
cos(2x) = cos^2(x) - sin^2(x)

Substitute this identity:
Y = 1 - cos^2(3x) - 6(cos^2(x) - sin^2(x)) + 2
Y = 1 - cos^2(3x) - 6cos^2(x) + 6sin^2(x) + 2

Finally, we can further simplify this expression by using the Pythagorean identity and other trigonometric identities as needed.