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?

Let $E=\sin(x)+\cos(x)$
Multiply and divide with $\sqrt2$. 
$$\begin{align}
E&=\sqrt2\left(\frac{\sin(x)}{\sqrt2} +\frac{\cos(x)}{\sqrt2}\right)\\ 
\implies E&= \sqrt2\sin\left(\frac \pi 4 + x\right)\\
\end{align}$$
Let $(\frac \pi4 + x)$ be $y$ then range of $y\in \left(\frac\pi4,\frac {3\pi}4\right)$ 
So $E= \sqrt2\sin(y)$.
Since max value of $\sin(y)$ is $1$ and min value is $\frac1{\sqrt2}$.
Therefore $\sqrt2 \ge E \ge 1$.
