To determine the value of [tex]\(\cos 45^\circ\)[/tex], we can refer to known trigonometric values associated with specific angles.
First, we know that [tex]\(\cos 45^\circ\)[/tex] is a well-known value in trigonometry. For a 45-degree angle, the value of the cosine function can be derived from the properties of a 45-45-90 triangle. In a 45-45-90 triangle, the lengths of the legs are equal, and the hypotenuse is [tex]\(\sqrt{2}\)[/tex] times the length of each leg.
Given this, the cosine of 45 degrees is the ratio of the adjacent side to the hypotenuse in a right triangle:
[tex]\[
\cos 45^\circ = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}
\][/tex]
Now we need to match this value with the provided options. The given options are:
A. 1
B. [tex]\(\frac{1}{2}\)[/tex]
C. [tex]\(\frac{\sqrt{2}}{2}\)[/tex]
From the calculation, we see that [tex]\(\cos 45^\circ = \frac{\sqrt{2}}{2}\)[/tex]. Hence, the correct answer is:
[tex]\[
\boxed{\text{C. } \frac{\sqrt{2}}{2}}
\][/tex]
