Question

Evaluate
$left parenthesis straight i right parenthesis open square brackets table row cell cos space straight theta end cell cell negative sin space straight theta space end cell row cell sin space straight theta end cell cell cos space straight theta end cell end table close square brackets space left parenthesis ii right parenthesis space open square brackets table row cell straight x squared minus straight x plus 1 end cell cell straight x minus 1 end cell row cell straight x plus 1 end cell cell straight x plus 1 end cell end table close square brackets$

Solution

left parenthesis straight i right parenthesis space Let space space increment space equals space open square brackets table row cell cos space straight theta end cell cell negative sin space straight theta space end cell row cell sin space straight theta end cell cell cos space straight theta end cell end table close square brackets space space equals space cos squared space straight theta space plus sin space squared space straight theta space equals 1 left parenthesis i i right parenthesis space Let space increment equals space space open square brackets table row cell straight x squared minus straight x plus 1 end cell cell straight x minus 1 end cell row cell straight x plus 1 end cell cell straight x plus 1 end cell end table close square brackets space equals space left parenthesis straight x squared space minus straight x plus 1 right parenthesis space left parenthesis straight x plus 1 right parenthesis minus left parenthesis straight x plus 1 right parenthesis space space space space space space space space space space space space space space space space space equals space space straight x to the power of 3 space end exponent minus straight x squared space plus straight x plus straight x squared space minus straight x plus 1 equals straight x to the power of 3 space space end exponent minus straight x squared space plus 2

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