Which of the following pairs of functions are identical?

For the following question, choose the correct answers from the codes $\left(A\right),\left(B\right),\left(C\right) \text{and} \left(D\right),$ defined as follows:
$\mathrm{f}$ is a function defined on the interval $\left[-\mathrm{1,1}\right]$ such that $\mathrm{f}\left(\sin 2x\right)=\mathrm{s}\mathrm{i}\mathrm{n}x+\mathrm{c}\mathrm{o}\mathrm{s}x$.

Statement I: If $x\in \left[-\frac{\text{π}}{4},\frac{\text{π}}{4}\right]$, then $\mathrm{f}\left({\tan }^{2}x\right)=\sec x.$

Statement II: $\mathrm{f}\left(x\right)=\sqrt{1+x},\forall x\in \left[-\mathrm{1,1}\right]$.

For the following questions, choose the correct answers from the codes $\left(a\right), \left(b\right), \left(c\right)$ and $\left(d\right)$ defined as follows:

Statement I: The range of $\mathrm{l}\mathrm{o}\mathrm{g}\left(\frac{1}{1+x^2}\right)$ is $\left(-\infty , \infty \right)$.

Statement II: When $0<x\le 1, \log x\in \left(-\infty , 0\right]$.