Solution:

Given expression is 4 (sin^4 30 cos^4 60) - 2/3 (sin^2 60 - cos^2 45) 1/2 tan^2 60.

First, we will simplify each term of the expression.

Simplification of the first term:

sin^4 30 can be written as (sin^2 30)^2 which is equal to (1/4)^2 = 1/16.

cos^4 60 can be written as (cos^2 60)^2 which is equal to (1/2)^2 = 1/4.

Therefore, sin^4 30 cos^4 60 = (1/16) * (1/4) = 1/64.

So, the first term of the expression becomes 4 * 1/64 = 1/16.

Simplification of the second term:

sin^2 60 can be written as (sin 60)^2 which is equal to (sqrt(3)/2)^2 = 3/4.

cos^2 45 can be written as (cos 45)^2 which is equal to (1/sqrt(2))^2 = 1/2.

Therefore, sin^2 60 - cos^2 45 = (3/4) - (1/2) = 1/4.

So, the second term of the expression becomes -(2/3) * (1/4) = -1/6.

Simplification of the third term:

tan^2 60 can be written as (sin^2 60)/(cos^2 60) which is equal to (3/4)/(1/4) = 3.

Therefore, 1/2 tan^2 60 = 1/2 * 3 = 3/2.

Now, we can substitute the simplified values of each term in the expression.

Final expression becomes:

1/16 - 1/6 + 3/2 = 23/24.

Therefore, the value of the given expression is 23/24.

4(sin^4 30 +cos^4 60) -2/3( sin^2 60 -cos^2 45) +1/2 tan^2 60. = 4[ (sin 30) 4 +(cos60)4]  -2/3[ (sin 60)2 -(cos 45)2]  +1/2(tan 60)2. = just substitute values of sin 30, cos 60, sin 60, cos 45, tan 60 = 1/2 answer. U substitute the values, u will get 1/2