By letting x = sin(pi/4 - y/2), the equation can be simplified to cos(y) = y using standard trig identities.
This has a unique solution y ~ 0.739 called the "Dottie number", but it seems like you can't do much better than approximate it numerically (though it may be interesting to note that there is a very simple algorithm to do this: just start with any real number and repeatedly apply the cosine function).
https://math.stackexchange.com/a/1174794 
