A) \[\frac{1}{2}(3\sqrt{5}+2\sqrt{10})\]
B) \[\frac{1}{2}(3\sqrt{5}-2\sqrt{10})\]
C) \[\frac{1}{2}(12+2\sqrt{50})\]
D) \[\frac{1}{2}(12-2\sqrt{50})\]
Correct Answer: C
Solution :
Given,\[{{\sinh }^{-1}}2+{{\sinh }^{-1}}3=x\] \[\Rightarrow \] \[\cosh ({{\sinh }^{-1}}2+{{\sinh }^{-1}}3)=\cosh x\] \[\Rightarrow \] \[\cosh ({{\sinh }^{-1}}2)\cosh ({{\sinh }^{-1}}3)\] \[+\sinh ({{\sinh }^{-1}}2)\sinh ({{\sinh }^{-1}}3)\] \[=\cosh x\] \[\Rightarrow \] \[\cosh x=\cosh ({{\cosh }^{-1}}\sqrt{1+{{2}^{2}}})\] \[\times \cosh ({{\cosh }^{-1}}\sqrt{1+{{3}^{2}}})+2\times 3\] \[\Rightarrow \] \[\cosh x=(\sqrt{5},\,\,\sqrt{10}+6)\times \frac{2}{2}\] \[=\frac{1}{2}(12+2\sqrt{50})\]
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