A) \[(\left. 0,\frac{{{\pi }^{2}}}{4} \right]\]
B) \[(0,\pi )\]
C) \[(0,2\pi ]\]
D) \[(0,1]\]
Correct Answer: B
Solution :
[a] \[y=(co{{t}^{-1}}x)(co{{t}^{-1}}(-x))\] \[={{\cot }^{-1}}(x)(\pi -co{{t}^{-1}}(x))\] Now \[{{\cot }^{-1}}(x)\] and \[(\pi -co{{t}^{-1}}(x))>0\] Using A.M\[\ge G.M.\]we get \[\frac{{{\cot }^{-1}}x+(\pi -co{{t}^{-1}}x)}{2}\ge \sqrt{(co{{t}^{-1}}x)(\pi -co{{t}^{-1}}x)}\] \[\Rightarrow 0<{{\cot }^{-1}}(x)(\pi -co{{t}^{-1}}(x))\] \[\le \left( \frac{{{\cot }^{-1}}x+(\pi -co{{t}^{-1}}x)}{2} \right)=\frac{{{\pi }^{2}}}{4}\] \[\Rightarrow 0<y\le \frac{{{\pi }^{2}}}{4}\]
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