Relations and Functions

Question

Find the multiplicative inverse of :

$space space space space space space space space space space space space straight z space equals space cos space straight theta space minus space 1 space sin space straight theta$

Answer

Multiplicative inverse of
$space space space space space space space space space space space space space space space straight z space equals space 1 over straight z space equals space fraction numerator 1 over denominator cos space straight theta space minus straight i space sin space straight theta end fraction space equals space fraction numerator 1 over denominator cosθ space minus space straight i space sin space straight theta end fraction. fraction numerator cos space straight theta space plus space straight i space sin space straight theta over denominator cos space straight theta space plus space straight i space sin space straight theta end fraction$

$space space space space space space space space space space space space space equals space space space fraction numerator cos space straight theta space plus space straight i space sin space straight theta over denominator cos squared space straight theta space minus space straight i squared space sin squared space straight theta end fraction space equals space fraction numerator cos space straight theta space plus space straight i space sin space straight theta over denominator cos squared space straight theta plus sin squared space straight theta end fraction equals fraction numerator cos space straight theta space plus space straight i space sin space straight theta over denominator 1 end fraction$

$space space space space space space space equals space cosθ plus space straight i space sinθ$