Show that (a/b+b/c+c/a)(a/c+c/b+b/a)=

1+(a/b+b/c)(b/c+c/a)(c/a+a/b)

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Show that (a/b+b/c+c/a)(a/c+c/b+b/a)=
1+(a/b+b/c)(b/c+c/a)(c/a+a/b)

Ayush Jha · Accepted Answer

We have to prove-

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

So LHS

$= (\frac{a}{b} + \frac{b}{c} + \frac{c}{a}) (\frac{a}{c} + \frac{c}{b} + \frac{b}{a}) = \frac{a^{2}}{b c} + \frac{a c}{b^{2}} + \frac{a b}{a b} + \frac{a b}{c^{2}} + \frac{b c}{b c} + \frac{b^{2}}{a c} + \frac{a c}{a c} + \frac{c^{2}}{a b} + \frac{b c}{a^{2}} = 1 + \frac{c}{a} (\frac{a}{c} + \frac{b^{2}}{c^{2}} + \frac{c}{b} + \frac{b}{a}) + \frac{a}{b} (\frac{a}{c} + \frac{b^{2}}{c^{2}} + \frac{c}{b} + \frac{b}{a}) = 1 + \frac{c}{a} (\frac{a b}{b c} + \frac{b^{2}}{c^{2}} + \frac{a c}{a b} + \frac{b c}{a c}) + \frac{a}{b} (\frac{a b}{b c} + \frac{b^{2}}{c^{2}} + \frac{a c}{a b} + \frac{b c}{a c}) = 1 + (\frac{a b}{b c} + \frac{b^{2}}{c^{2}} + \frac{a c}{a b} + \frac{b c}{a c}) (\frac{c}{a} + \frac{a}{b}) = 1 + (\frac{a}{b} + \frac{b}{c}) (\frac{b}{c} + \frac{c}{a}) (\frac{c}{a} + \frac{a}{b}) = R H S$

Hence proved

Show that (a/b+b/c+c/a)(a/c+c/b+b/a)= 1+(a/b+b/c)(b/c+c/a)(c/a+a/b)

Show that (a/b+b/c+c/a)(a/c+c/b+b/a)=

1+(a/b+b/c)(b/c+c/a)(c/a+a/b)