To solve the problem [tex]\(\sin^{-1}\left(\tan\left(\frac{\pi}{4}\right)\right)\)[/tex], we need to evaluate it step-by-step.
First, let's evaluate the inner function [tex]\(\tan\left(\frac{\pi}{4}\right)\)[/tex].
Recall that:
[tex]\[
\tan\left(\frac{\pi}{4}\right) = 1
\][/tex]
Next, we need to evaluate the inverse sine (arcsine) of [tex]\(1\)[/tex]:
[tex]\[
\sin^{-1}(1)
\][/tex]
By definition, [tex]\(\sin^{-1}(x)\)[/tex] is the value [tex]\(y\)[/tex] such that [tex]\(\sin(y) = x\)[/tex] and [tex]\(y\)[/tex] lies in the range [tex]\([- \frac{\pi}{2}, \frac{\pi}{2}]\)[/tex].
The value of [tex]\(y\)[/tex] that satisfies [tex]\(\sin(y) = 1\)[/tex] within this range is:
[tex]\[
\frac{\pi}{2}
\][/tex]
Thus:
[tex]\[
\sin^{-1}(1) = \frac{\pi}{2}
\][/tex]
Therefore:
[tex]\[
\sin^{-1}\left(\tan\left(\frac{\pi}{4}\right)\right) = \frac{\pi}{2}
\][/tex]
Given the possible options:
[tex]\[
0, \frac{\pi}{2}, \pi, 2\pi
\][/tex]
The correct one is:
[tex]\[
\boxed{\frac{\pi}{2}}
\][/tex]
