Question:
Let $\mathrm{M}$ and $\mathrm{m}$ respectively be the maximum and minimum values of the function $f(x)=\tan ^{-1}(\sin x+\cos x)$ in $\left[0, \frac{\pi}{2}\right]$, Then the value of $\tan (\mathrm{M}-\mathrm{m})$ is equal to:
Correct Option: , 4
Solution:
Let $g(x)=\sin x+\cos x=\sqrt{2} \sin \left(x+\frac{\pi}{4}\right)$
$\mathrm{g}(\mathrm{x}) \in[1, \sqrt{2}]$ for $\mathrm{x} \in[0, \pi / 2]$
$f(x)=\tan ^{-1}(\sin x+\cos x) \in\left[\frac{\pi}{4}, \tan ^{-1} \sqrt{2}\right]$
$\tan \left(\tan ^{-1} \sqrt{2}-\frac{\pi}{4}\right)=\frac{\sqrt{2}-1}{1+\sqrt{2}} \times \frac{\sqrt{2}-1}{\sqrt{2}-1}=3-2 \sqrt{2}$