If abc = 1 then prove that (1/(1 + a + b^(−1) )) + (1/(1 + b + c^(−1) )) + (1/(1 + c + a^(−1) )) = 1


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Question Number 206322 by MATHEMATICSAM last updated on 12/Apr/24

$If a b c = 1 then prove that$ $\frac{1}{1 + a + b^{- 1}} + \frac{1}{1 + b + c^{- 1}} + \frac{1}{1 + c + a^{- 1}} = 1$
Commented by Rasheed.Sindhi last updated on 12/Apr/24

$F o r s i m i l a r q u e s t i o n s e e$ $You can't use 'macro parameter character #' in math mode$
	Answered by som(math1967) last updated on 12/Apr/24

	$\frac{b}{b + a b + 1} + \frac{1}{1 + b + a b} + \frac{a b}{a b + a b c + b}$ $[a b c = 1 ∴ a b = c^{- 1}]$ $= \frac{b}{b + 1 + a b} + \frac{1}{b + 1 + a b} + \frac{a b}{b + 1 + a b}$ $= \frac{b + 1 + a b}{b + 1 + a b} = 1$
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