If A + B + C = π. Find value of

$open vertical bar table row cell sin space left parenthesis straight A plus straight B plus straight C right parenthesis end cell cell space sin space straight B end cell cell space space cos space straight C end cell row cell negative sin space straight B end cell 0 cell space tan space straight A end cell row cell cos left parenthesis straight A plus straight B right parenthesis end cell cell space minus tan space straight A end cell cell space 0 end cell end table close vertical bar$

Solution

space Let space increment space equals space open vertical bar table row cell sin space left parenthesis straight A plus straight B plus straight C right parenthesis end cell cell space sin space straight B end cell cell space space cos space straight C end cell row cell negative sin space straight B end cell 0 cell space tan space straight A end cell row cell cos left parenthesis straight A plus straight B right parenthesis end cell cell space minus tan space straight A end cell cell space 0 end cell end table close vertical bar space equals open vertical bar table row 0 cell space space sin space straight B end cell cell space space cos space straight C end cell row cell negative sin space straight B end cell 0 cell space space tan space straight A end cell row cell negative cos space straight C end cell cell space space minus tan space straight A end cell 0 end table close vertical bar space space space space space space space space space space space space space open square brackets table row cell because space sin space left parenthesis straight A plus straight B plus straight C right parenthesis equals negative sin space straight pi space equals 0 comma space cos space left parenthesis straight A plus straight B right parenthesis equals cos left parenthesis straight pi minus straight C right parenthesis equals negative cos space straight C end cell end table close square brackets space space space space space space space space space space equals space 0 space space space space space space space space space space space space space space space space open square brackets table row cell because space determinant space of space straight a space skew space symmetric space matrix space of space odd space order space is space always space zero end cell end table close square brackets space

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