MCQ
$4\, {\tan ^{ - 1}}\frac{1}{5} - {\tan ^{ - 1}}\frac{1}{{70}} + {\tan ^{ - 1}}\frac{1}{{99}} = $
- A$\frac{\pi }{2}$
- B$\frac{\pi }{3}$
- ✓$\frac{\pi }{4}$
- DNone of these
$ = 2\, {\tan ^{ - 1}}\left[ {\frac{{\frac{2}{5}}}{{1 - \frac{1}{{25}}}}} \right] - {\tan ^{ - 1}}\frac{1}{{70}} + {\tan ^{ - 1}}\frac{1}{{99}}$
$ = 2\, {\tan ^{ - 1}}\left( {\frac{5}{{12}}} \right) - {\tan ^{ - 1}}\frac{1}{{70}} + {\tan ^{ - 1}}\frac{1}{{99}}$
$ = {\tan ^{ - 1}}\left[ {\frac{{\frac{5}{6}}}{{1 - \frac{{25}}{{144}}}}} \right] - {\tan ^{ - 1}}\frac{1}{{70}} + {\tan ^{ - 1}}\frac{1}{{99}}$
$ = {\tan ^{ - 1}}\left( {\frac{{120}}{{119}}} \right) - {\tan ^{ - 1}}\frac{1}{{70}} + {\tan ^{ - 1}}\frac{1}{{99}}$
$ = {\tan ^{ - 1}}\left( {\frac{{120}}{{119}}} \right) + {\tan ^{ - 1}}\left[ {\frac{{\frac{1}{{99}} - \frac{1}{{70}}}}{{1 + \frac{1}{{99}}.\frac{1}{{70}}}}} \right]$
$ = {\tan ^{ - 1}}\left( {\frac{{120}}{{119}}} \right) + {\tan ^{ - 1}}\left( {\frac{{ - 29}}{{6931}}} \right)$
$ = {\tan ^{ - 1}}\frac{{120}}{{119}} - {\tan ^{ - 1}}\frac{{29}}{{6931}} = {\tan ^{ - 1}}\frac{{120}}{{119}} - {\tan ^{ - 1}}\frac{1}{{239}}$
$ = {\tan ^{ - 1}}\left[ {\frac{{\frac{{120}}{{119}} - \frac{1}{{239}}}}{{1 + \frac{{120}}{{119}} \times \frac{1}{{239}}}}} \right] $
$= {\tan ^{ - 1}}(1) = \frac{\pi }{4}$.
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where $[x]$ denotes the greatest integer less than or equal to $x$. Let $f \circ:(-1,1) \rightarrow R$ be the composite function defined by $(f \circ g)(x)=f(g(x))$. Suppose $c$ is the number of points in the interval $(-1,1)$ at which $f \circ g$ is NOT continuous, and suppose $d$ is the number of points in the interval $(-1,1)$ at which $f \circ g$ is $NOT$ differentiable. Then the value of $c+d$ is. . . . .