91Ó°ÊÓ

The angle given is π/24, which can be broken down into half angles. Since the half-angle identities require π/2 multiples or related angles, we need to utilize the double-angle identity to write: $$\cos \frac{\pi}{24} = \cos \frac{1}{2} \cdot \frac{\pi}{12}$$ Now that the angle is broken down into a half angle, we can proceed to use the half-angle identities.

The half-angle identity for cosine is given by: $$\cos \frac{\theta}{2} = \pm \sqrt{\frac{1 + \cos\theta}{2}}$$ We need to determine the correct sign of the result. In this case, we have: $$\cos \frac{\pi}{24} = \cos \frac{1}{2} \cdot \frac{\pi}{12}$$ Since the angle (Ï€/12) lies in the first quadrant (where cosine is positive), we will use the positive square root. Applying the half-angle identity for cosine to the given expression, we get: $$\cos \frac{\pi}{24} = \sqrt{\frac{1 + \cos \frac{\pi}{12}}{2}}$$

Notice that we can simplify the cosine of π/12 using the angle addition identity. To do this, we express π/12 as the sum of two known angles, say π/4 and π/6. Then: $$\cos \frac{\pi}{12} = \cos \left(\frac{\pi}{4} + \frac{\pi}{6}\right)$$ Now apply the angle addition formula for cosine: $$\cos (\alpha + \beta) = \cos \alpha \cdot \cos \beta - \sin \alpha \cdot \sin \beta$$ Substitute the angles into the formula: $$\cos \frac{\pi}{12} = \cos \frac{\pi}{4} \cdot \cos \frac{\pi}{6} - \sin \frac{\pi}{4} \cdot \sin \frac{\pi}{6}$$ We know the values for cos(π/4), cos(π/6), sin(π/4), and sin(π/6): $$\cos \frac{\pi}{12} = \left(\frac{\sqrt{2}}{2}\right) \cdot \left(\frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2}\right) \cdot \left(\frac{1}{2}\right)$$

Now, we plug the simplified expression for cos(Ï€/12) into the half-angle formula: $$\cos \frac{\pi}{24} = \sqrt{\frac{1 + \left(\frac{\sqrt{6} + \sqrt{2}}{4}\right)}{2}}$$ To simplify the expression inside the square root further, find a common denominator and add the fractions: $$\cos \frac{\pi}{24} = \sqrt{\frac{4 + \sqrt{6} + \sqrt{2}}{8}}$$ Finally, simplify the expression by taking the square root: $$\cos \frac{\pi}{24} = \frac{\sqrt{4 + \sqrt{6} + \sqrt{2}}}{2\sqrt{2}}$$ Thus, the exact value of the given expression, cos(Ï€/24), is: $$\cos \frac{\pi}{24} = \frac{\sqrt{4 + \sqrt{6} + \sqrt{2}}}{2\sqrt{2}}$$