MCQ
If $\tan ^{ - 1}x + \tan ^{ - 1}y + \tan ^{ - 1}z = \frac{\pi }{2},$ then
- A$x + y + z - xyz = 0$
- B$x + y + z + xyz = 0$
- C$xy + yz + zx + 1 = 0$
- ✓$xy + yz + zx - 1 = 0$
$ \Rightarrow \,\,{\tan ^{ - 1}}\,\left[ {\frac{{x + y + z - xyz}}{{1 - xy - yz - xz}}} \right] = \frac{\pi }{2}$
$ \Rightarrow \,\,\left[ {\frac{{x + y + z - xyz}}{{1 - xy - yz - zx}}} \right] = \tan \frac{\pi }{2} = \frac{1}{0}$
Hence $xy + yz + zx - 1 = 0$.
Trick : $x = y = z = \frac{1}{{\sqrt 3 }},$ so that
${\tan ^{ - 1}}\frac{1}{{\sqrt 3 }} + {\tan ^{ - 1}}\frac{1}{{\sqrt 3 }} + {\tan ^{ - 1}}\frac{1}{{\sqrt 3 }} = \frac{\pi }{2}$
Obviously $ (d)$ holds for these values of $x, y, z.$
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Statement $-1 :$ $gof $ is differentiable at $x=0$ and its derivative is continuous at that point
Statement $-2 :$ $gof $ is twice differentiable at $x=0 $