using the identity sin^2(x) + cos^2(x) = 1, we can rewrite the expression as:

28/sin^2(31) + sin^2(59)
= 28/cos^2(59) + sin^2(59)
= 28/(1 - sin^2(59)) + sin^2(59)

Now, we can substitute sin^2(59) with x:

28/(1-x) + x

Now, we need to find the value of x:

sin^2(59) = sin^2(90-31) = sin^2(90)cos^2(31) = 1 * cos^2(31) = cos^2(31)

Therefore, x = cos^2(31)

Now, we can substitute x back into the expression:

28/(1 - cos^2(31)) + cos^2(31)

Using the identity sin^2(x) + cos^2(x) = 1, we know that:

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

So the expression becomes:

28/sin^2(31) + sin^2(31)

= 28*sin^2(31) + sin^2(31)

= 29*sin^2(31)

Therefore, the final answer is 29*sin^2(31).