Which of the following are identical functions? [Note: [x] denotes greatest integer less than or equal to x, x denotes fractional part of x and operatornamesgn x dentoes signum function of x respectively.]

Question

Which of the following are identical functions? [Note: [x] denotes greatest integer less than or equal to x, x denotes fractional part of x and operatornamesgn x dentoes signum function of x respectively.] (A) f(x)= sin 2[ x ]+ cos 2[ x ] (B) g(x)= operatornamecosec2 x - cot 2 x  (C) h(x)= operatornamesgn((x2+1/x2)) (D) k ( x )= ln e [ operatornamesgn( e x + e - x )]

Tardigrade Admin · Accepted Answer

text For (A) (D) x ∈ R text For (B) x ∈ R-I text For (C) x ∈ R-0 y=1 text for all

Q. Which of the following are identical functions?
[Note : $[x]$ denotes greatest integer less than or equal to $x, {x}$ denotes fractional part of $x$ and $sgn x$ dentoes signum function of $x$ respectively.]

$f (x) = sin^{2} [{x}] + cos^{2} [{x}]$

$g (x) = cosec^{2} {x} - cot^{2} {x}$

$h (x) = sgn (\frac{x ^{2} + 1}{x ^{2}})$

$k (x) = ln e^{[sgn (e^{x} + e^{- x})]}$

$For (A)$ & $(D) x \in R$
$For (B) x \in R - I$
$For (C) x \in R - 0$
$y = 1 for all$

Q. Which of the following are identical functions? [Note : [x] denotes greatest integer less than or equal to x,{x} denotes fractional part of x and sgnx dentoes signum function of x respectively.]

f(x)=sin2[{x}]+cos2[{x}]

g(x)=cosec2{x}−cot2{x}

h(x)=sgn(x2x2+1​)

k(x)=lne[sgn(ex+e−x)]