$(1)cosh(arcsinh(\frac{1}{2}))$
$cosh^{2}(arcsinh(\frac{1}{2}))=1+sinh^{2}(arcsinh (\frac{1}{2}))$
$cosh(arcsinh(\frac{1}{2}))=\sqrt{1+(\frac{1}{2})^ {2}}=\frac{\sqrt{5}}{2}$
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$(2) sinh(2arcsinh(\frac{1}{2}))$
$=2sinh(arcsinh(\frac{1}{2}))cosh(arcsinh(\frac{1} {2}))$
$=1.\frac{\sqrt{5}}{2}=\frac{\sqrt{5}}{2}$
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