Using the sum and difference of cosines formula, find the exact value of cos(π/12).

Steps to Solve

1. Identify Angles for Cosine Formula

We can express $\frac{\pi}{12}$ as the difference of two angles whose cosine values are known. We can use: $$\frac{\pi}{12} = \frac{\pi}{3} - \frac{\pi}{4}$$

2. Apply the Cosine Difference Formula

Using the cosine difference formula: $$\cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b)$$

Substituting $a = \frac{\pi}{3}$ and $b = \frac{\pi}{4}$: $$\cos\left(\frac{\pi}{12}\right) = \cos\left(\frac{\pi}{3}\right)\cos\left(\frac{\pi}{4}\right) + \sin\left(\frac{\pi}{3}\right)\sin\left(\frac{\pi}{4}\right)$$

3. Calculate Cosine and Sine Values

Next, we evaluate these trigonometric values:

• $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$
• $\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$
• $\sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2}$
• $\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$

Substituting these values into the equation: $$\cos\left(\frac{\pi}{12}\right) = \left(\frac{1}{2}\right)\left(\frac{\sqrt{2}}{2}\right) + \left(\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{2}}{2}\right)$$

4. Simplify the Expression

Calculating the values:

• First term: $\frac{1 \cdot \sqrt{2}}{2 \cdot 2} = \frac{\sqrt{2}}{4}$
• Second term: $\frac{\sqrt{3} \cdot \sqrt{2}}{2 \cdot 2} = \frac{\sqrt{6}}{4}$

Combining both terms: $$\cos\left(\frac{\pi}{12}\right) = \frac{\sqrt{2}}{4} + \frac{\sqrt{6}}{4} = \frac{\sqrt{2} + \sqrt{6}}{4}$$

5. Final Result

Thus, we find: $$\cos\left(\frac{\pi}{12}\right) = \frac{\sqrt{2} + \sqrt{6}}{4}$$