Find the exact value of \( \cos 45^{\circ} \) \( \cos 45^{\circ}=\square \) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

To find the exact value of \( \cos 45^{\circ} \), we can use the properties of a 45-45-90 right triangle. **Step 1: Understand the 45-45-90 Triangle** A 45-45-90 triangle is an isosceles right triangle where both non-right angles are \(45^{\circ}\). If the legs of the triangle each have length 1, the hypotenuse can be found using the Pythagorean theorem: \[ \text{Hypotenuse} = \sqrt{1^2 + 1^2} = \sqrt{2} \] **Step 2: Apply the Definition of Cosine** Cosine of an angle in a right triangle is the ratio of the length of the adjacent side to the hypotenuse. For \(45^{\circ}\): \[ \cos 45^{\circ} = \frac{\text{Adjacent side}}{\text{Hypotenuse}} = \frac{1}{\sqrt{2}} \] **Step 3: Rationalize the Denominator** To present the answer in its simplest form, rationalize the denominator by multiplying the numerator and the denominator by \( \sqrt{2} \): \[ \cos 45^{\circ} = \frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2} \] **Final Answer:** \[ \cos 45^{\circ} = \frac{\sqrt{2}}{2} \]