Question

Prove that:
$open vertical bar table row straight alpha cell space space straight beta end cell cell space space space space straight gamma end cell row cell straight alpha squared end cell cell space space straight beta squared end cell cell space space space straight gamma squared end cell row cell straight beta plus straight gamma end cell cell space space space straight gamma plus straight alpha end cell cell space space space space straight alpha plus straight beta end cell end table close vertical bar space equals space left parenthesis straight beta minus straight gamma right parenthesis space left parenthesis straight gamma minus straight alpha right parenthesis thin space left parenthesis straight alpha minus straight beta right parenthesis thin space left parenthesis straight alpha plus straight beta plus straight gamma right parenthesis$

Solution

Let space increment space equals space open vertical bar table row straight alpha cell space straight beta end cell cell space space straight gamma end cell row cell straight alpha squared end cell cell space straight beta squared end cell cell space space straight gamma squared end cell row cell straight beta plus straight gamma end cell cell space straight gamma plus straight alpha end cell cell space space straight alpha plus straight beta end cell end table close vertical bar

space open vertical bar table row straight alpha cell space space straight beta end cell cell space space space straight gamma end cell row cell straight alpha squared end cell cell space space straight beta squared end cell cell space space space straight gamma squared end cell row cell straight alpha plus straight beta plus straight gamma end cell cell space space straight alpha plus straight beta plus straight gamma end cell cell space space space straight alpha plus straight beta plus straight gamma end cell end table close vertical bar comma space by space straight R subscript 3 plus straight R subscript 1 equals space left parenthesis straight alpha plus straight beta plus straight gamma right parenthesis space open vertical bar table row straight alpha cell space space straight beta end cell cell space space straight gamma end cell row cell straight alpha squared end cell cell space space straight beta squared end cell cell space space straight gamma squared end cell row 1 1 1 end table close vertical bar equals space left parenthesis straight alpha plus straight beta plus straight gamma right parenthesis space open vertical bar table row 1 cell space 1 end cell cell space space 1 end cell row straight alpha cell space straight beta end cell cell space straight gamma end cell row cell straight alpha squared end cell cell space straight beta squared end cell cell space space straight gamma squared end cell end table close vertical bar equals space left parenthesis straight alpha plus straight beta plus straight gamma right parenthesis space open vertical bar table row 1 0 0 row straight alpha cell straight beta minus straight alpha end cell cell straight gamma minus straight alpha end cell row cell straight alpha squared end cell cell straight beta squared minus straight alpha squared end cell cell straight gamma squared minus straight alpha squared end cell end table close vertical bar comma space by space straight C subscript 2 minus straight C subscript 1 comma space space space straight C subscript 3 minus straight C subscript 1 equals space left parenthesis straight alpha plus straight beta plus straight gamma right parenthesis space left parenthesis straight beta minus straight alpha right parenthesis space left parenthesis straight gamma minus straight alpha right parenthesis space open vertical bar table row 1 cell space 0 end cell cell space space 0 end cell row straight alpha cell space 1 end cell cell space 1 end cell row cell straight alpha squared end cell cell space space straight beta plus straight alpha end cell cell space space straight gamma plus straight alpha end cell end table close vertical bar equals space left parenthesis straight alpha plus straight beta plus straight gamma right parenthesis space left parenthesis straight beta minus straight alpha right parenthesis space left parenthesis straight gamma minus straight alpha right parenthesis space open vertical bar table row 1 cell space space space space 1 end cell row cell straight beta plus straight alpha end cell cell space space space space straight gamma plus straight alpha end cell end table close vertical bar equals space left parenthesis straight alpha plus straight beta plus straight gamma right parenthesis space left parenthesis straight beta minus straight alpha right parenthesis space left parenthesis straight gamma minus straight alpha right parenthesis thin space left parenthesis straight gamma plus straight alpha minus straight beta minus straight alpha right parenthesis equals space left parenthesis straight alpha plus straight beta plus straight gamma right parenthesis thin space left parenthesis straight beta minus straight alpha right parenthesis space left parenthesis straight gamma minus straight alpha right parenthesis thin space left parenthesis straight gamma minus straight beta right parenthesis equals space left parenthesis straight alpha plus straight beta plus straight gamma right parenthesis thin space left parenthesis straight beta minus straight gamma right parenthesis thin space left parenthesis straight gamma minus straight alpha right parenthesis thin space left parenthesis straight alpha minus straight beta right parenthesis.

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