2022/025) if $a-\frac{1}{a} = b$, $b-\frac{1}{b} = c$, $c-\frac{1}{c} = a$ evaluate $\frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca}$

 We are given $a-\frac{1}{a} = b\cdots(1)$

$b-\frac{1}{b} = c \cdots(2)$

$c-\frac{1}{c} = a\cdots(3)$

From (1)$\frac{1}{a} = a-b $

dividing both sides by b

$\frac{1}{ab}  =  \frac{a}{b} - 1  = a(b-c)  -1 $ using |(2)

similarly

$\frac{1}{bc}  =  \frac{b}{c} - 1  = b(c-a)  -1 $

and $\frac{1}{ca} = \frac{c}{a} - 1 = -c(a-b) = 1 $

adding above 3 we have $\frac{1}{ab} + \frac{1}{bc} + \frac{1}{ca}= - 3 $