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:

Question

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: (A) ie - e - i z (B) ie - e i z (C) - ie - e - i z  (D) None of these

Tardigrade Admin · Accepted Answer

( f - ig )( z )= f ( z )- ig ( z )= sin z - i cos z =- i ( cos z + i sin z )=- i e i z=θ (say) Now (f + ig ) *( f - ig )( z )=( f + ig )( f - ig )( z ) =( f + ig )(θ)= f (θ)+ ig (θ) = sin θ+ i cos θ= i ( cos θ- i sin θ) = i e - i θ= i e - i (- ie i z)= ie - e i z

Q. Let $f (z) = sin z$ and $g (z) = cos z$ . If $$ denotes a composition of functions, then the value of $(f + i g) (f - i g)$ is:

$i e^{- e^{- i z}}$

$i e^{- e^{i z}}$

$- i e^{- e^{- i z}}$

None of these

Q. Let f(z)=sinz and g(z)=cosz. If ∗ denotes a composition of functions, then the value of (f+ig)∗(f−ig) is:

ie−e−iz

ie−eiz

−ie−e−iz

None of these

(f−ig)(z)=f(z)−ig(z)=sinz−icosz =−i(cosz+isinz)=−ieiz=θ (say) Now (f +ig)∗(f−ig)(z)=(f+ig)(f−ig)(z) =(f+ig)(θ)=f(θ)+ig(θ) =sinθ+icosθ=i(cosθ−isinθ) =ie−iθ=ie−i(−ieiz)=ie−eiz

Q. Let $f (z) = sin z$ and $g (z) = cos z$ . If $$ denotes a composition of functions, then the value of $(f + i g) (f - i g)$ is:

$i e^{- e^{- i z}}$

$i e^{- e^{i z}}$

$- i e^{- e^{- i z}}$