Which pair(s) of function(s) is/are equal? (w | vedclass

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(D) For two functions $f(x)$ and $g(x)$ to be equal,they must have the same domain and $f(x) = g(x)$ for all $x$ in the domain.Check option $A$: Let $

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(D) For two functions $f(x)$ and $g(x)$ to be equal,they must have the same domain and $f(x) = g(x)$ for all $x$ in the domain.Check option $A$: Let $x = 	an 	heta$. Then $f(x) = \cos(2	heta) = \frac{1 - 	an^2 	heta}{1 + 	an^2 	heta} = \frac{1 - x^2}{1 + x^2} = g(x)$. Both are defined for all $x \in \mathbb{R}$. Thus,$A$ is equal.Check option $B$: Let $x = \cot 	heta$. Then $g(x) = \sin(2	heta) = \frac{2\cot 	heta}{1 + \cot^2 	heta} = \frac{2x}{1 + x^2} = f(x)$. Both are defined for all $x \in \mathbb{R}$. Thus,$B$ is equal.Check option $C$: For $f(x)$,$\cot^{-1} x > 0$ for all $x \in \mathbb{R}$,so $\operatorname{sgn}(\cot^{-1} x) = 1$. Thus $f(x) = e^{\ln(1)} = 1$. For $g(x)$,$[1 + \{x\}] = 1$ because $0 \le \{x\} Since all pairs are equal,the correct option is $D$.

Which pair$(s)$ of function$(s)$ is/are equal? (where ${x}$ and $[x]$ denote the fractional part and integral part functions respectively.)

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