Which of the following are identical functions? (where [ x ] denotes greatest integer less than or equal x , x denotes fractional part of x and operatornamesgn x denotes signum function of x respectively.)

Question

Which of the following are identical functions? (where [ x ] denotes greatest integer less than or equal x , x denotes fractional part of x and operatornamesgn x denotes signum function of x respectively.) (A) f ( x )= operatornamesgn(| x |+1) (B) g(x)= sin 2( ln x)+ cos 2( ln x) (C) h(x)=(2/π)( sin -1 x + cos -1 x ) (D) k ( x )= sec 2[ x ]- tan 2 [ x ]

Tardigrade Admin · Accepted Answer

(A) operatornamesgn(|x|+1)=1 ∀ x ∈ R. (B) sin 2( ln x )+ cos 2( ln x )=1 ∀ x ∈ R + (C) (2/π)( sin -1 x + cos -1 x )=1 ∀ x ∈ R. (D) sec 2[ x ]- tan 2 [x] = sec 2 0°- tan 2 0°=1 ∀ x ∈ R.

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

$f (x) = sgn (∣ x ∣ + 1)$

$g (x) = sin^{2} (ln x) + cos^{2} (ln x)$

$h (x) = \frac{2}{π} (sin^{- 1} {x} + cos^{- 1} {x})$

$k (x) = sec^{2} [{x}] - tan^{2} {[x]}$

(A) $sgn (∣ x ∣ + 1) = 1\forall x \in R$ .
(B) $sin^{2} (ln x) + cos^{2} (ln x) = 1\forall x \in R^{+}$
(C) $\frac{2}{π} (sin^{- 1} {x} + cos^{- 1} {x}) = 1\forall x \in R$ .
(D) $sec^{2} [{x}] - tan^{2} {[x]} = sec^{2} 0^{\circ} - tan^{2} 0^{\circ} = 1\forall x \in R$ .

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

f(x)=sgn(∣x∣+1)

g(x)=sin2(lnx)+cos2(lnx)

h(x)=π2​(sin−1{x}+cos−1{x})

k(x)=sec2[{x}]−tan2{[x]}