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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