To simplify the expression:

cos75° can be expressed as cos(45° + 30°), so we can apply the sum-to-product identity for cosine to get:

cos75° = cos(45° + 30°) = cos45°cos30° - sin45°sin30°
= (√2/2)(√3/2) - (√2/2)(1/2)
= (√6 - √2) / 4

Similarly, cos15° can be expressed as cos(45° - 30°), so we can apply the sum-to-product identity for cosine to get:

cos15° = cos(45° - 30°) = cos45°cos30° + sin45°sin30°
= (√2/2)(√3/2) + (√2/2)(1/2)
= (√6 + √2) / 4

Now, substitute these values into the expression:

√3( cos75° - cos15°) / (1 - 2sin^2 15°)
= √3((√6 - √2)/4 - (√6 + √2)/4) / (1 - 2sin^2 15°)
= √3((-2√2)/4) / (1 - 2sin^2 15°)
= -√6 / (1 - 2sin^2 15°)

Now, we need to find the value of sin15° to simplify the expression further. Using the half-angle identity for sine:

sin15° = √((1 - cos30°) / 2)
= √((1 - √3/2) / 2)
= √((2 - √3) / 4)

Substitute this into the expression:

-√6 / (1 - 2((2 - √3) / 4)^2)
= -√6 / (1 - (2(4 - 4√3 + 3) / 16))
= -√6 / (1 - (8 - 8√3 + 6) / 16)
= -√6 / (1 - (14 - 8√3) / 16)
= -√6 / (1 - (14 - 8√3) / 16)
= -√6 / (2 + 8√3) / 16)
= -√6 / ((16 + 128√3) / 16)
= -√6 * 16 / (16 + 128√3)
= -16√6 / (16 + 128√3)

Therefore, the simplified expression is -16√6 / (16 + 128√3)