evaluate. write your answer in simplified, rationalized form. do not round. cos 45° =

Explanation:

Step1: Recall cosine value of 45 - degree angle

In a right - isosceles triangle with angles 45 - 45 - 90, if the legs have length $a$, the hypotenuse has length $c=\sqrt{a^{2}+a^{2}}=\sqrt{2a^{2}}=\sqrt{2}a$ by the Pythagorean theorem. The cosine of an angle in a right triangle is defined as $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. For $\theta = 45^{\circ}$, the adjacent side to the 45 - degree angle and the hypotenuse are related as $\cos45^{\circ}=\frac{a}{\sqrt{2}a}$.

Step2: Simplify the expression

$\frac{a}{\sqrt{2}a}=\frac{1}{\sqrt{2}}$. Rationalize the denominator by multiplying the numerator and denominator by $\sqrt{2}$: $\frac{1\times\sqrt{2}}{\sqrt{2}\times\sqrt{2}}=\frac{\sqrt{2}}{2}$.

Answer:

$\frac{\sqrt{2}}{2}$