To determine [tex]\(\tan 45^\circ\)[/tex], we start by considering the definition of the tangent function, which is the ratio of the opposite side to the adjacent side in a right-angled triangle.
The angle [tex]\(45^\circ\)[/tex] often appears in an isosceles right triangle. In such a triangle, where the two non-hypotenuse sides are equal, the following ratio holds:
[tex]\[ \tan 45^\circ = \frac{\text{opposite}}{\text{adjacent}} \][/tex]
For an isosceles right triangle with a [tex]\(45^\circ\)[/tex] angle, let's denote both equal sides as having a length of 1. In this case, the ratio becomes:
[tex]\[ \tan 45^\circ = \frac{1}{1} = 1 \][/tex]
Thus, the value of [tex]\(\tan 45^\circ\)[/tex] is exactly 1.
Given the multiple-choice options:
A. [tex]\(\sqrt{2}\)[/tex]
B. [tex]\(\frac{1}{\sqrt{2}}\)[/tex]
C. [tex]\(\frac{1}{2}\)[/tex]
D. [tex]\(1\)[/tex]
The correct answer is:
[tex]\[ \boxed{1} \][/tex]
