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Step 1: Understand the Function
We are given the function
$f(x) = \tan^{-1}(\sin x + \cos x)$
over the interval $[0, \frac{\pi}{2}]$. We want to find the maximum (M) and minimum (m) of $f(x)$ on this interval and then compute $\tan(M - m)$.
Step 2: Simplify and Find the Range of $\sin x + \cos x$
First, note that
$\sin x + \cos x = \sqrt{2}\,\sin\!\Bigl(x + \frac{\pi}{4}\Bigr).$
For $x \in \Bigl[0, \frac{\pi}{2}\Bigr]$, the expression
$x + \frac{\pi}{4}$
ranges from
$\frac{\pi}{4}$
to
$\frac{3\pi}{4}.$
Hence
$\sin\Bigl(x + \frac{\pi}{4}\Bigr)$
takes values in
$[\frac{\sqrt{2}}{2},\, \frac{\sqrt{2}}{2}\cdot\sqrt{2}] = [\frac{\sqrt{2}}{2},1].$
Therefore,
$\sin x + \cos x$
ranges from
$\sqrt{2}\cdot \frac{\sqrt{2}}{2} = 1$
to
$\sqrt{2} \cdot 1 = \sqrt{2}.$
So,
$\sin x + \cos x \in [1, \sqrt{2}].
Step 3: Find the Range of $f(x)$
Since $f(x) = \tan^{-1}(\sin x + \cos x)$ and $\sin x + \cos x \in [1, \sqrt{2}]$,
$f(x)$ will range from
$\tan^{-1}(1)$
to
$\tan^{-1}(\sqrt{2}).$
We know:
$\tan^{-1}(1) = \frac{\pi}{4}.$
$\tan^{-1}(\sqrt{2})$ is a positive acute angle.
Thus,
$m = \frac{\pi}{4}$
and
$M = \tan^{-1}(\sqrt{2}).
Step 4: Determine $\tan(M - m)$
We want
$\tan\bigl(\tan^{-1}(\sqrt{2}) - \frac{\pi}{4}\bigr).$
Let $a = \tan^{-1}(\sqrt{2})$ and $b = \frac{\pi}{4}.$ Then
$\tan(a) = \sqrt{2}$
and
$\tan(b) = 1.$
Using the tangent difference formula:
$\tan(a - b)
= \frac{\tan a - \tan b}{1 + \tan a \cdot \tan b}
= \frac{\sqrt{2} - 1}{1 + \sqrt{2}\cdot 1}
= \frac{\sqrt{2} - 1}{1 + \sqrt{2}}.
Step 5: Simplify the Result
To simplify $\frac{\sqrt{2} - 1}{1 + \sqrt{2}},$ multiply numerator and denominator by $(\sqrt{2} - 1)$:
$\frac{\sqrt{2} - 1}{1 + \sqrt{2}}
\times \frac{\sqrt{2} - 1}{\sqrt{2} - 1}
= \frac{(\sqrt{2} - 1)^2}{(1 + \sqrt{2})(\sqrt{2} - 1)}.
Expand the numerator:
$(\sqrt{2} - 1)^2 = 2 - 2\sqrt{2} + 1 = 3 - 2\sqrt{2}.$
Expand the denominator:
$(1 + \sqrt{2})(\sqrt{2} - 1) = 1 \cdot (\sqrt{2} - 1) + \sqrt{2} \cdot (\sqrt{2} - 1) = (\sqrt{2} - 1) + (2 - \sqrt{2}) = 1.
Therefore:
$\tan(M - m)
= \frac{3 - 2\sqrt{2}}{1}
= 3 - 2\sqrt{2}.
Final Answer
$3 - 2\sqrt{2}.
