To find the limit of the given expression as x approaches 2, we can apply the following steps:

lim (x->2) (cos(x))^1/(sin(2x))^2

As x approaches 2:

(cos(2))^1 / (sin(4))^2

Using trigonometric identities:

(cos(2))^1 / (sin(2*2))^2
(cos(2))^1 / (sin(4))^2

Again, using trigonometric identities:

sin(4) = 2sin(2)cos(2)

Therefore:

(cos(2))^1 / (2sin(2)cos(2))^2
(cos(2))^1 / (4sin^2(2)cos^2(2))

And since:

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

Substitute this into our expression:

(cos(2))^1 / (4sin^2(2)(1 - sin^2(2)))

= cos(2) / (4sin^2(2) - 4sin^4(2))

As x approaches 2, cos(2) is cos(2), sin(2) is sin(2).

Hence, after taking the limit:

cos(2) / (4sin^2(2) - 4sin^4(2)) = cos(2) / (4sin^2(2) - 4sin^4(2))

This is the limit of the given expression as x approaches 2.