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 write $$\sin x + \cos x = \sqrt 2 \left(\sin x \cos (\pi/4) + \cos x\sin(\pi/4)\right) = \sqrt 2 \sin(x+\pi/4)$$ now use the fact that 
$\frac{ \sqrt 2} 2 \le \sin(x + \pi/4) \le 1 \text{ for } 0 \le x \le \pi/2$ to conclude $$1 \le \sin x + \cos x  \le \sqrt 2 \text{ for } 0 \le x \le \pi/2.$$
