Prove that $\sin(x) + \cos(x) \geq 1$ $\forall x\in[0,\pi/2]: \sin{x}+\cos{x} \ge 1.$
I am really bad at trigonometric functions, how could I prove it?

you can consider the function $f(x) = \sin{x} +\cos{x} -1$ then see that in $[0,\pi /4)$ ,
$f'(x) >0$ 
and in $(\pi/4, \pi/2], f'(x) <0$ 
now see what happens at $0$ and $\pi/2$
