Given:
sinθ - cosθ = (√6 - √2)/2

We need to find the value of:
4(sin^3θ - cos^3θ)^2

Solution:
To find the value of 4(sin^3θ - cos^3θ)^2, let's first simplify sinθ - cosθ using the trigonometric identity:

sin(θ - φ) = sinθcosφ - cosθsinφ

Let φ = π/4, then:

sin(θ - π/4) = sinθcos(π/4) - cosθsin(π/4)
sin(θ - π/4) = (1/√2)(sinθ - cosθ) [1]

Using the given equation sinθ - cosθ = (√6 - √2)/2, we substitute this value into equation [1]:

sin(θ - π/4) = (1/√2)((√6 - √2)/2)
sin(θ - π/4) = (√6 - √2)/4√2
sin(θ - π/4) = (√6 - √2)/(4√2) [2]

Now, let's expand (sin^3θ - cos^3θ)^2 using the identity:

(a - b)^2 = a^2 - 2ab + b^2

Substituting sinθ - cosθ into a and b:

(sinθ - cosθ)^2 = sin^2θ - 2sinθcosθ + cos^2θ
(sinθ - cosθ)^2 = sin^2θ + cos^2θ - 2sinθcosθ
(sinθ - cosθ)^2 = 1 - 2sinθcosθ [3]

Using the identity sin(θ - φ) = (√6 - √2)/(4√2) obtained in equation [2], we substitute this value into equation [3]:

1 - 2sinθcosθ = 1 - 2(√6 - √2)/(4√2)
1 - 2sinθcosθ = 1 - (√6 - √2)/(2√2)
1 - 2sinθcosθ = (2√2 - √6 + √2)/(2√2)
1 - 2sinθcosθ = (√2 - √6)/(2√2) [4]

Finally, substitute equation [4] into the expression 4(sin^3θ - cos^3θ)^2:

4(sin^3θ - cos^3θ)^2 = 4(1 - 2sinθcosθ)^2
4(sin^3θ - cos^3θ)^2 = 4((√2 - √6)/(2√2))^2
4(sin^3θ - cos^3θ)^2 = 4(√2 - √6)^2/(2√2)^2
4(sin^3θ - cos^3θ)^2 = 4(2 - 2√12 + 6)/(8)
4(sin^3θ - cos^3θ)^2 = (2 - √12 + 3)/