Function $f(x) =\tan^{–1} (\sin x + \cos x)$ is monotonic increasing when

$0 < x<\frac{π}{4}$

$f'(x)=\frac{\cos x-\sin x}{1+(\sin x+\cos x)^2}$

$f(x)$ is monotonic increasing when $f'(x)>0$

$⇒\frac{\cos x-\sin x}{1+(\sin x+\cos x)^2}>0⇒\cos x-\sin x>0$

$⇒\sqrt{2}\cos(x+\frac{π}{4})>0$

$⇒\frac{-π}{2}<x+\frac{π}{4}<\frac{π}{2}$ (∵ $\cos θ$ is positive when $\frac{-π}{2}<θ<\frac{π}{2}$)