Find the exact value of \( \cos 345^{\circ} \).

To find the exact value of \( \cos 345^\circ \), we can use the cosine subtraction formula. We can express \( 345^\circ \) as \( 360^\circ - 15^\circ \). Thus, we have: \[ \cos 345^\circ = \cos(360^\circ - 15^\circ) \] Using the cosine subtraction identity, we know that: \[ \cos(360^\circ - \theta) = \cos \theta \] Therefore, we can simplify: \[ \cos 345^\circ = \cos 15^\circ \] Next, we need to find \( \cos 15^\circ \). We can use the cosine subtraction formula again, expressing \( 15^\circ \) as \( 45^\circ - 30^\circ \): \[ \cos 15^\circ = \cos(45^\circ - 30^\circ) \] Using the cosine subtraction formula: \[ \cos(a - b) = \cos a \cos b + \sin a \sin b \] we have: \[ \cos 15^\circ = \cos 45^\circ \cos 30^\circ + \sin 45^\circ \sin 30^\circ \] Now, we substitute the known values: - \( \cos 45^\circ = \frac{\sqrt{2}}{2} \) - \( \cos 30^\circ = \frac{\sqrt{3}}{2} \) - \( \sin 45^\circ = \frac{\sqrt{2}}{2} \) - \( \sin 30^\circ = \frac{1}{2} \) Substituting these values into the equation gives: \[ \cos 15^\circ = \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right) \left(\frac{1}{2}\right) \] Calculating each term: \[ \cos 15^\circ = \frac{\sqrt{2} \cdot \sqrt{3}}{4} + \frac{\sqrt{2}}{4} \] Combining the terms: \[ \cos 15^\circ = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4} \] Thus, we find: \[ \cos 345^\circ = \cos 15^\circ = \frac{\sqrt{6} + \sqrt{2}}{4} \] Therefore, the exact value of \( \cos 345^\circ \) is: \[ \boxed{\frac{\sqrt{6} + \sqrt{2}}{4}} \]