MCQ
$\sin ({\cot ^{ - 1}}x) =$
- A$\sqrt {1 + {x^2}} $
- B$x$
- C${(1 + {x^2})^{ - 3/2}}$
- ✓${(1 + {x^2})^{ - 1/2}}$
Now, $\cos ec \theta =\sqrt {1+ {\cot ^2 \theta}} = \sqrt {{x^2} +1} $.
$\therefore \,\,\,\sin \theta = \frac{1}{{\cos ec\,\theta }} = \frac{1}{{\sqrt {1 + {x^2}} }}\,\, \Rightarrow \,\theta = {\sin ^{ - 1}}\frac{1}{{\sqrt {1 + {x^2}} }}$
Hence $\sin \,({\cot ^{ - 1}}x)\, = \sin \,\left( {{{\sin }^{ - 1}}\frac{1}{{\sqrt {1 + {x^2}} }}} \right)$
$ = \frac{1}{{\sqrt {1 + {x^2}} }} = {(1 + {x^2})^{ - 1/2}}$
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($S1$) $f(x)=0$ for only one value of $x$ is $[0, \pi]$.
($S2$) $\mathrm{f}(\mathrm{x})$ is decreasing in $\left[0, \frac{\pi}{2}\right]$ and increasing in $\left[\frac{\pi}{2}, \pi\right] .$