Inverse Trigonometric Functions

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Question

Evaluate :

$open vertical bar table row cell 2 space cos space straight theta end cell cell negative 2 space sin space straight theta end cell row cell sin space straight theta end cell cell cos space straight theta end cell end table close vertical bar$

Solution

Let space space space space space space space space space space space space space space space space space space increment space equals space open vertical bar table row cell 2 space cos space straight theta end cell cell negative 2 space sin space straight theta end cell row cell sin space straight theta end cell cell cos space straight theta end cell end table close vertical bar space space space space space space space space space space space space space space space space space space space space space space space space space space equals space left parenthesis 2 space cos space straight theta right parenthesis space left parenthesis cos space straight theta right parenthesis space minus space left parenthesis negative space 2 space sin space straight theta right parenthesis space left parenthesis sin space straight theta right parenthesis equals space 2 space cos squared space straight theta space space plus space 2 space sin squared space straight theta space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space 2 space left parenthesis cos squared straight theta space plus space sin squared space straight theta right parenthesis equals 2 left parenthesis 1 right parenthesis equals 2 space space space space space