[tex] \frac{x + y + z}{ {x}^{ - 1} {y}^{ - 1} + {y}^{ - 1} {z}^{ - 1} + {z}^{ - 1} {x}^{ - 1} } = xyz[/tex]
prove the following
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[tex] \frac{x + y + z}{ {x}^{ - 1} {y}^{ - 1} + {y}^{ - 1} {z}^{ - 1} + {z}^{ - 1} {x}^{ - 1} } = xyz[/tex]
prove the following
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Step-by-step explanation:
[tex] \huge \underline{ \orange{To \: Prove:}}[/tex]
[tex] \bold \red{ \frac{ x + y + z }{ \blue{ {x}^{ - 1 } {y}^{ - 1} + {y}^{ - 1} {z}^{ - 1} + {z}^{ - 1} {x}^{ - 1} } } } = \green{xyz}[/tex]
[tex] \huge \underline{ \gray{ Proof:}}[/tex]
[tex] \bold\pink{L.H.S.:}[/tex]
[tex] \bold \red{ \frac{x + y + z}{ \blue{ {x}^{ - 1} {y}^{ - 1} + {y}^{ - 1} {z}^{ - 1} + {z}^{ - 1} {x}^{ - 1} }} } \\ = \frac{ \bold \red{x + y + z}}{ \bold \blue{ \frac{1}{xy} + \frac{1}{yz} + \frac{1}{zx} }} \\ = \frac{ \bold \red{x + y + z}}{ \bold \blue{ \frac{z + x + y}{xyz} }} \\ = \frac{ \bold \red{(x + y + z)\blue{(xyz)}}}{ \bold \blue{(x + y + z)}} \\ = \bold \green {xyz}[/tex]
[tex] \bold \pink{R.H.S.:}[/tex]
[tex] \bold \green{xyz}[/tex]
[tex] \: \: \ \: \: \: \: \: \: \bold \pink{ { \huge∴} \: \: L.H.S.= R.H.S.}[/tex]
[tex] \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \tt \green{ \underline { \orange{hence \: \: proved.}}}[/tex]