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Q.
Let $f ( z )=\sin z$ and $g ( z )=\cos z$. If $*$ denotes a composition of functions, then the value of $( f + ig ) *( f - ig )$ is:
Relations and Functions - Part 2
Solution:
$( f - ig )( z )= f ( z )- ig ( z )=\sin \,z - i\, \cos \,z$
$=- i (\cos \,z + i\,sin \,z )=- i e ^{ i z}=\theta$ (say)
Now (f $+ ig ) *( f - ig )( z )=( f + ig )( f - ig )( z )$
$=( f + ig )(\theta)= f (\theta)+ ig (\theta)$
$=\sin \,\theta+ i\, \cos \,\theta= i (\cos\, \theta- i \,\sin\, \theta)$
$= i e ^{- i\, \theta}= i e ^{- i \left(- ie ^{ i z}\right)}= ie ^{- e ^{ i z}}$