Extrema of $(1+\sin x)(1+\cos x)$ 
Find the extrema of $(1+\sin x)(1+\cos x)$ without using calculus.

I was able to figure out the minima by observing that each of the brackets range from $0$ to $2$. Therefore the minima has to be $0$ when either one of the brackets is zero.
However I couldnt figure out the maxima. I tried expanding it to complete the square but it didnt quite work out well.
Any hint is appreciated!

We can also use the inequality
\begin{gather*}
AM\geq GM\\
So,\ \\
\sqrt{( 1+\sin x)( 1+\cos x)} \leq \frac{1+\sin x+1+\cos x}{2}\\
Now,\ the\ function\ on\ the\ RHS\ is\\
1+\frac{\sin x+\cos x}{2} =1+\frac{1}{\sqrt{2}}\sin\left( x+\frac{\pi }{4}\right)\\
The\ maxima\ of\ this\ function\ is\ 1+\frac{1}{\sqrt{2}} ,\\
because\ maximum\ value\ of\ \sin\left( x+\frac{\pi }{4}\right) =1\\
\sqrt{( 1+\sin x)( 1+\cos x)} \leq 1+\frac{1}{\sqrt{2}}\\
( 1+\sin x)( 1+\cos x) \leq \left( 1+\frac{1}{\sqrt{2}}\right)^{2} =\frac{3+2\sqrt{2}}{2} =\frac{3}{2} +\sqrt{2}
\end{gather*}
