Let $\theta$ be an arbitrary acute angle. Prove that $\sin (\theta) + \cos(\theta) \ge 1$.
$$\big(\sin (\theta) + \cos (\theta)\big)^2 = 1 + 2 \sin(\theta)\cos(\theta)\ge 0$$
so, \begin{align*}\big(\sin(\theta)) + \cos(\theta)\big)^2 &> 1\\
\big(\sin(\theta)+ \cos(\theta)\big)^2 &\ge 1\end{align*}
Best Answer
For a geometric approach: Draw a right triangle with hypotenuse $1$. Mark one of the acute angles as $\theta$. Then $\sin(\theta)+\cos(\theta)$ is just the sum of the two legs, which is greater than the length of the hypotenuse.