Question

$Simplify space cosθ space open square brackets table row cell cos space straight theta end cell cell sin space straight theta end cell row cell negative sin space straight theta end cell cell cos space straight theta end cell end table close square brackets plus sin space straight theta space open square brackets table row cell sin space straight theta end cell cell negative cos space straight theta end cell row cell cos space straight theta end cell cell sin space straight theta end cell end table close square brackets space space space space space space space space space space space space space space space space space space space$

Solution

Consider
$open square brackets table row cell cos space straight theta end cell cell sin space straight theta end cell row cell negative sin space straight theta end cell cell cos space straight theta end cell end table close square brackets plus sin space straight theta space open square brackets table row cell sin space straight theta end cell cell negative cos space straight theta end cell row cell cos space straight theta end cell cell sin space straight theta end cell end table close square brackets space space space space space space space space space space space space space space space space space space space space equals open square brackets table row cell cos squared space straight theta end cell cell sin space straight theta space cos space straight theta end cell row cell negative sin space straight theta space cos squared space straight theta end cell cell cos squared space straight theta end cell end table close square brackets plus open square brackets table row cell sin squared space straight theta end cell cell negative sin space straight theta space cos space straight theta end cell row cell sin space straight theta space cos space straight theta end cell cell sin squared space straight theta end cell end table close square brackets space space space space space space space space space space space space space space space space space space space equals space open square brackets table row cell cos squared space straight theta space plus sin squared space straight theta end cell cell sin space straight theta space cos space straight theta space minus space sin space straight theta space cos space straight theta end cell row cell negative sin space straight theta space cos space straight theta space plus sin space straight theta space cos space straight theta end cell cell cos squared space straight theta space plus space sin squared space straight theta end cell end table close square brackets space space space space space space space space space space space space space space space space space space space equals space open square brackets table row 1 0 row 0 1 end table close square brackets$

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