The set of all possible values of θ in the interval (0, π) for...

for lying on the same side $(\sin \theta+\cos \theta-1)(1+2-1)>0$$\begin{aligned} & \Rightarrow \sin \theta+\cos \theta>1 \\ & \Rightarrow \frac{1}{\sqrt{2}} \sin \theta+\frac{1}{\sqrt{2}} \cos \theta>\frac{1}{\sqrt{2}} \end{aligned}$$$<img src="/api/proxy-image?url=https$$\begin{aligned} & \Rightarrow \sin \left(\theta+\frac{\pi}{4}\right)>\frac{1}{\sqrt{2}} \\ & \Rightarrow \frac{\pi}{4} < \theta+\frac{\pi}{4} < \frac{3 \pi}{4} \\ & \Rightarrow 0 < \theta < \frac{\pi}{2}\end{aligned}$$