Question

Show that $f\left(x\right)={\tan }^{-1}(\sin x+\cos x)$ is an increasing function in $(0,\frac{\pi }{4})$.

Answer 1

We have,

$f\left(x\right)={\tan }^{-1}(\sin x+\cos x)$

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

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

For $x\in (0,\frac{\pi }{4}),\cos x>\sin x$

$\frac{df}{dx}>0$

$f\left(x\right)$ increases in $(0,\frac{\pi }{4})$.