Prove that the determinant

$open vertical bar table row straight x cell space space space space sin space straight theta end cell cell space space space space space cos space straight theta end cell row cell negative sin space straight theta end cell cell negative straight x end cell 1 row cell cos space straight theta end cell cell space space 1 end cell straight x end table close vertical bar$ is independent of ө.

Solution

Let space space space space increment space space equals space open vertical bar table row straight x cell space space space space sin space straight theta end cell cell space space space space space cos space straight theta end cell row cell negative sin space straight theta end cell cell negative straight x end cell 1 row cell cos space straight theta end cell cell space space 1 end cell straight x end table close vertical bar space space space space space space space space space space space space space space space equals straight x space space open vertical bar table row cell negative straight x end cell cell space space space 1 end cell row 1 cell space space space straight x end cell end table close vertical bar space minus sin space straight theta space open vertical bar table row cell negative sin space straight theta end cell cell space space 1 end cell row cell cos space straight theta end cell cell space space straight x end cell end table close vertical bar plus cos space straight theta space open vertical bar table row cell negative sin space straight theta end cell cell space minus straight x end cell row cell cos space straight theta end cell cell space space space space 1 end cell end table close vertical bar space space space space space space space space space space space space space space space equals straight x left parenthesis negative straight x squared minus 1 right parenthesis space minus sin space left parenthesis negative straight x space sin space straight theta space minus cos space straight theta right parenthesis minus cos space straight theta left parenthesis negative sin space straight theta plus cos space straight theta right parenthesis space space space space space space space space space space space space space space space equals negative straight x cubed minus straight x plus straight x space sin squared space straight theta space plus sin space straight theta space cos space straight theta space minus sin space straight theta space cos space straight theta space plus straight x space cos squared space straight theta space space space space space space space space space space space space space space space equals space minus straight x cubed minus straight x plus straight x left parenthesis sin squared space straight theta space plus cos squared space straight theta right parenthesis space equals space straight x cubed minus straight x space space space space space space space space space space space space space space space equals negative straight x cubed space comma space which space is space independent space of space space straight theta

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