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Question
Show that: (a − b)(a + b) + (b − c)(b + c) + (c − a)( c + a) = 0
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Solution
\[\text { LHS } = \left( a - b \right)\left( a + b \right) + \left( b - c \right)\left( b + c \right) + \left( c + a \right)\left( c - a \right)\]
\[ = a^2 - b^2 + b^2 - c^2 + c^2 - a^2 \left[ \because \left( a + b \right)\left( a - b \right) = a^2 - b^2 \right]\]
\[ = a^2 - b^2 + b^2 - c^2 + c^2 - a^2 \]
\[ = 0\]
= RHS
Because LHS is equal to RHS, the given equation is verified.
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