To determine the value of [tex]\(\cos 45^{\circ}\)[/tex], we can use our knowledge of trigonometric functions and special angles.
The angle [tex]\(45^{\circ}\)[/tex] is known as one of the special angles in trigonometry, and there are well-known values for trigonometric functions at these angles.
For an angle of [tex]\(45^{\circ}\)[/tex]:
[tex]\[
\cos 45^{\circ} = \frac{1}{\sqrt{2}}
\][/tex]
However, it is customary to rationalize the denominator when expressing trigonometric values. Thus, by multiplying the numerator and the denominator by [tex]\(\sqrt{2}\)[/tex], we get:
[tex]\[
\cos 45^{\circ} = \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} \approx 0.7071
\][/tex]
The numerical approximation confirms that [tex]\( \cos 45^{\circ} \approx 0.7071 \)[/tex].
From the provided options:
A. [tex]\(\frac{1}{\sqrt{2}}\)[/tex]
B. [tex]\(\sqrt{2}\)[/tex]
C. [tex]\(\frac{1}{2}\)[/tex]
D. 1
Option A matches with our derived value before rationalizing the denominator:
[tex]\[
\cos 45^{\circ} = \frac{1}{\sqrt{2}}
\][/tex]
Thus, the correct answer is:
A. [tex]\(\frac{1}{\sqrt{2}}\)[/tex].
