I just need a solution verification, because I thought the answer I got was too odd
For $\cos x > 0$
$$\cos x + \cos x \ge 1$$
$$\cos x \ge \frac 12$$
$$x\in [0,\frac{\pi}{3}]$$
$\cos x <0$ can be ignored because it won’t satisfy
$$\sin x +\cos x > 1$$
$$\sin (x+\frac{\pi}{4})> \frac{1}{\sqrt 2}$$
$$x+\frac{\pi}{4}>\frac{\pi}{4} \text {or} \frac{3\pi}{4}$$
$$x>0$$
Then $$x\in (0,\frac{\pi}{3})$$
Again, I don’t think I got this right. It seems very loose and has a lot of missing parts. Please let me know what I did wrong.