Answer
$\epsilon = \frac{1}{K + 1 }$
Work Step by Step
We know the efficiency of an ideal heat engine equation is
$\epsilon = \frac{|W|}{|Q_H|} $
And the equation of coefficient of performance (COP) in a reversible refrigerator is
$K = \frac{|Q_H| - |W|}{|W|} $
$K = \frac{|Q_H| }{|W|} - \frac{|W|}{|W|} $
$K = \frac{|Q_H| }{|W|} - 1$ ,
but $\frac{|W|}{|Q_H|} = \epsilon $. substitute $\epsilon$ into the $K$ equation
$K = \frac{1}{\epsilon} - 1$ , rearrange the equation
$\frac{1}{\epsilon} = K + 1 $
$\epsilon = \frac{1}{K + 1 }$
