<s>[INST] Find the greatest and least values of $(\sin^{-1}x)^2+(\cos^{-1}x)^2$ Find the upper and lower limit of 

$$
(\sin^{-1}x)^2+(\cos^{-1}x)^2
$$

My Attempt:
$$
\frac{-\pi}{2}\leq\sin^{-1}x\leq \frac{\pi}{2}\quad\&\quad0\leq\cos^{-1}x\leq\pi\\(\sin^{-1}x)^2\leq\frac{\pi^2}{4}\quad\&\quad(\cos^{-1}x)^2\leq\pi^2\\
0\leq(\sin^{-1}x)^2+(\cos^{-1}x)^2\leq\frac{\pi^2}{4}+\pi^2=\frac{5\pi^2}{4}
$$
Here, I can see the upper limit is $\frac{5\pi^2}{4}$ which is fine. But, $0$ is one lower limit not the lower limit.
Why am I not getting the lower limit in my approach ?
How do I approach similar problems involving max and min, when you don't get the lower or upper limits ?
 [/INST] Let $\sin^{-1}x=a,\cos^{-1}x=b$
$a+b=\dfrac\pi2$
$$a^2+b^2=\left(\dfrac\pi2-b\right)^2+b^2=\dfrac{\pi^2}4+2\left(b-\dfrac\pi4\right)^2-2\left(\dfrac\pi4\right)^2$$
Again , $0\le b\le\pi\implies0\le\left(b-\dfrac\pi4\right)^2\le\left(\pi-\dfrac\pi4\right)^2$
</s>