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!

$(1 + \sin x)(1+ \cos x)=1+\sin x + \cos x + \sin x \cos x.$
Note that $\sin x \cos x = \frac 12 \sin 2x.$
Note also that $\sin x + \cos x = \sqrt 2 \left ( \frac {\sqrt 2}{2} \sin x + \frac {\sqrt 2}{2} \cos x \right) = \sqrt 2 \left ( \cos \frac {\pi}{4} \sin x + \sin \frac{\pi}{4} \cos x \right )= \sqrt 2 \sin (x+ \frac{\pi}{4}).$
Both of these factors reach their maximum when $x=\frac{\pi}{4}$, so the maximum is $\frac 32 + \sqrt 2$.
