Maximum and Minimum value of an inverse function 
Find the maximum and minimum value of
  $\arcsin \left(x\right)^3+\arccos \left(x\right)^3$.
given that $-1\le x\le 1$

I have solved the problem but i am just curious to know if there are any other ways to solve this particular problem other than the method i used below.
By using the fact that $\arcsin \left(x\right)+\arccos \left(x\right)$ =$\frac{\pi }{2}$
i found that
$\arcsin \left(x\right)^3+\arccos \left(x\right)^3$=$3\left(\frac{\pi }{2}\right)^2\left(\left\{\arcsin \left(x\right)-\frac{\pi }{4}\right\}^2+\frac{\left(8\pi -3\pi ^2\right)}{48}\right)$
so it is minimum when $\left\{\arcsin \left(x\right)-\frac{\pi }{4}\right\}^2$$=0$
or $x=\sin \left(\frac{\pi }{4}\right)$
Therefore minimum value$=\frac{1}{32}\pi ^3$
and it is maximum when $\arcsin \left(x\right)=-\frac{\pi }{2}$
Therefore maximum value=$\frac{7}{8}\pi ^3$

Help:
a^3+b^3=(a+b)((a+b)^2-3ab)
Let p=Pi
ASinx+ACosx=p/2
f(x)=(ASinx)^3+(ACosx)^3
      =p/2 (p^2/4-3.ASinx.ACosx)
      =p/2 (p^2/4-3.(p/2).ASinx+3.(ASinx)^2)
This shall be maximum for ASinx= -(p/2)
Hence max value of the function will be:
f (x)max=(7/8) (p^2)
