The number sqrt(2)/2 comes up in several places . To make the typing easier I will replace it with .707 and then restore at the end. I am going to limit my analysis to The standard domain restriction on sin-1 . This restricts x to [-.5 , .5] and generates angles in the first and second quadrant. Further, I will take cos-1(.707) = pi/4 (radian measure) and sin(pi/4) = cos(pi/4) = .707 (first quadrant values)
With all of this, the expression is sin( pi/4 -sin-1(2 x)) This is the sine of a difference, so the trig angle difference formula can be used. The result is sin(pi/4) cos(sin-1(2 x)) - cos(pi/4) sin(sin-1(2 x)) . This evaluates to
.707 [ cos(sin-1(2 x) - sin(sin-1(2x)) ].
With the domain restriction mentioned above sin( sin-1( 2 x)) = 2 x. cos( sin-1(2 x) is a bit more complicated, but a standard triangle analysis show that it is sqrt(1 - 4 x2) So the expression becomes
.707[ sqrt( 1 - 4 x2) - 2 x ] restoring sqrt(2)/2 gives
[sqrt(2)/2] [sqrt(1 - 4 x2) - 2 x] valid for -.5 < x < .5
The graph of this expression is interesting. It is clear that it is the top half of a closed curve. Presumable, the rest of the closed curve could be obtained by properly removing the domain restriction.