A) 1
B) 2
C) 3
D) No solution
Correct Answer: A
Solution :
\[\tan \,({{\cos }^{-1}}x)=\sin \,\,\left( {{\cos }^{-1}}\frac{1}{2} \right)\] |
\[{{\cot }^{-1}}\frac{1}{2}=t\] \[t\in \left( 0,\,\,\frac{\pi }{2} \right)\] |
\[\operatorname{cott}=\frac{1}{2}\] |
\[\sin t=\frac{2}{\sqrt{5}}\] |
\[\tan \,({{\cos }^{-1}}x)\] |
\[{{\cos }^{-1}}x=t\in [0,\,\,\pi ]\] |
\[\cos t=x.\] |
\[\tan =\sqrt{1-{{x}^{2}}}\] \[t\in \left[ 0,\,\,\frac{\pi }{2} \right]\] |
\[\sqrt{1-{{x}^{2}}}=\frac{2}{\sqrt{5}}\] |
\[1-{{x}^{2}}=\frac{4}{5}\] \[\Rightarrow \] \[{{x}^{2}}=\frac{1}{\sqrt{5}}\] |
\[x=\pm \frac{1}{\sqrt{5}}\] |
\[x=\frac{1}{\sqrt{5}}\] only one solution. |
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