Question

Show that:  (a − b)(a + b) + (b − c)(b + c) + (c − a)( c + a) = 0

Answer 1

\[\text {  LHS } = \left( a - b \right)\left( a + b \right) + \left( b - c \right)\left( b + c \right) + \left( c + a \right)\left( c - a \right)\]

\[ = a^2 - b^2 + b^2 - c^2 + c^2 - a^2 \left[ \because \left( a + b \right)\left( a - b \right) = a^2 - b^2 \right]\]

\[ = a^2 - b^2 + b^2 - c^2 + c^2 - a^2 \]

\[ = 0\]

 = RHS

Because LHS is equal to RHS, the given equation is verified.