Use a sum or difference formula to find the exact value of the trigonometric function. \( \cos \frac{\pi}{12} \) \( \cos \frac{\pi}{12}=\square \) (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

To find the exact value of \( \cos \frac{\pi}{12} \), we can use the cosine difference formula. The angle \( \frac{\pi}{12} \) can be expressed as a difference of two angles whose cosine values we know. We can write: \[ \frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4} \] Now, we can apply the cosine difference formula: \[ \cos(a - b) = \cos a \cos b + \sin a \sin b \] where \( a = \frac{\pi}{3} \) and \( b = \frac{\pi}{4} \). Now we will calculate \( \cos \frac{\pi}{3} \), \( \sin \frac{\pi}{3} \), \( \cos \frac{\pi}{4} \), and \( \sin \frac{\pi}{4} \): - \( \cos \frac{\pi}{3} = \frac{1}{2} \) - \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \) - \( \cos \frac{\pi}{4} = \frac{\sqrt{2}}{2} \) - \( \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2} \) Now substituting these values into the cosine difference formula: \[ \cos \frac{\pi}{12} = \cos \frac{\pi}{3} \cos \frac{\pi}{4} + \sin \frac{\pi}{3} \sin \frac{\pi}{4} \] \[ = \left(\frac{1}{2}\right) \left(\frac{\sqrt{2}}{2}\right) + \left(\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{2}}{2}\right) \] \[ = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} \] \[ = \frac{\sqrt{2} + \sqrt{6}}{4} \] Thus, the exact value of \( \cos \frac{\pi}{12} \) is: \[ \cos \frac{\pi}{12} = \frac{\sqrt{2} + \sqrt{6}}{4} \]