Prove the following identities:
$open vertical bar table row 1 cell space space space 1 end cell cell space space space 1 end cell row straight a cell space space straight b end cell cell space space space straight c end cell row cell straight a cubed end cell cell space space space straight b cubed end cell cell space space space straight c cubed end cell end table close vertical bar space equals space left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis straight a plus straight b plus straight c right parenthesis$

Solution

$Let space increment space equals space open vertical bar table row 1 cell space space space straight a end cell cell space space straight a cubed end cell row 1 cell space space straight b end cell cell space space straight b cubed end cell row 1 cell space space straight c end cell cell space space straight c cubed end cell end table close vertical bar equals space open vertical bar table row 1 cell space space 1 end cell cell space space 1 end cell row straight a cell space space straight b end cell cell space space straight c end cell row cell straight a cubed end cell cell space space straight b cubed end cell cell space space straight c cubed end cell end table close vertical bar comma space by space interchanging space row space and space columns equals space open vertical bar table row 1 0 cell space space 0 end cell row straight a cell space space straight b minus straight a end cell cell space space straight c minus straight a end cell row cell straight a cubed end cell cell space space straight b cubed minus straight a cubed end cell cell space space straight c cubed minus straight a cubed end cell end table close vertical bar comma space by space straight C subscript 2 minus straight C subscript 1 comma space space straight C subscript 3 minus straight C subscript 1 equals space open vertical bar table row cell straight b minus straight a end cell cell space space space space space space straight c minus straight a end cell row cell straight b cubed minus straight a cubed end cell cell space space space space space straight c cubed minus straight a cubed end cell end table close vertical bar comma space expanding space with space first space row. equals space open vertical bar table row cell straight b minus straight a end cell cell space space space space space space space space space space space space space space space space space space straight c minus straight a end cell row cell left parenthesis straight b minus straight a right parenthesis thin space left parenthesis straight b squared plus straight a squared plus ba right parenthesis end cell cell space space space space space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis straight c squared plus straight a squared plus ca right parenthesis end cell end table close vertical bar equals space left parenthesis straight b minus straight a right parenthesis thin space left parenthesis straight c minus straight a right parenthesis space open vertical bar table row 1 cell space space space space space space space space space 1 end cell row cell straight b squared plus straight a squared plus ba end cell cell space space space space space space space straight c squared plus straight a squared plus ca end cell end table close vertical bar equals space left parenthesis straight b minus straight a right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space left square bracket straight c squared plus straight a squared plus ca minus straight b squared minus straight a squared minus ba right square bracket equals space left parenthesis straight b minus straight a right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space left square bracket left parenthesis straight c squared minus straight b squared right parenthesis space plus space left parenthesis ca minus ba right parenthesis right square bracket equals space left parenthesis straight b minus straight a right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space left square bracket left parenthesis straight c minus straight b right parenthesis thin space left parenthesis straight c plus straight b right parenthesis space plus space straight a space left parenthesis straight c minus straight b right parenthesis right square bracket equals space left parenthesis straight b minus straight a right parenthesis thin space left parenthesis straight c minus straight a right parenthesis thin space left parenthesis straight c minus straight b right parenthesis thin space left parenthesis straight c plus straight b plus straight a right parenthesis space equals space left parenthesis straight a minus straight b right parenthesis thin space left parenthesis straight b minus straight c right parenthesis thin space left parenthesis straight c minus straight a right parenthesis space left parenthesis straight a plus straight b plus straight c right parenthesis.$

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