It's hard to follow all of the various brackets and parentheses. But, if I'm reading it correctly, there is a numerical answer.
Put x = d/2 and y = √(4 - d²)/2. Then x² + y² = 1, and the equation reduces to a more manageable form.
arcsin(y) - y√(1 - y²) = π/4, if d > 0 or
arcsin(y) + y√(1 - y²) = π/4, if d < 0.
The equations have the solutions (obtained numerically)
y ≈ 0.914771017573, d > 0 and y ≈ 0.40397275330, d < 0.
Plugging these into the equation 2y = √(4 - d²) produces the solutions
d ≈ 0.8079455066, if d > 0
d ≈ -1.829542035, if d < 0.
Put x = d/2 and y = √(4 - d²)/2. Then x² + y² = 1, and the equation reduces to a more manageable form.
arcsin(y) - y√(1 - y²) = π/4, if d > 0 or
arcsin(y) + y√(1 - y²) = π/4, if d < 0.
The equations have the solutions (obtained numerically)
y ≈ 0.914771017573, d > 0 and y ≈ 0.40397275330, d < 0.
Plugging these into the equation 2y = √(4 - d²) produces the solutions
d ≈ 0.8079455066, if d > 0
d ≈ -1.829542035, if d < 0.