To find the exact value of $\cos 15^\circ$ using a sum or difference formula, we can proceed as follows:

Step 1: Identify the Formula

The angle $15^\circ$ can be expressed as: $15^\circ = 45^\circ - 30^\circ$ Thus, we can use the cosine difference formula: $\cos (a - b) = \cos a \cos b + \sin a \sin b$

Step 2: Substitute Known Values

Using $a = 45^\circ$ and $b = 30^\circ$: $\cos 15^\circ = \cos 45^\circ \cos 30^\circ + \sin 45^\circ \sin 30^\circ$

We know:

• $\cos 45^\circ = \frac{\sqrt{2}}{2}$
• $\cos 30^\circ = \frac{\sqrt{3}}{2}$
• $\sin 45^\circ = \frac{\sqrt{2}}{2}$
• $\sin 30^\circ = \frac{1}{2}$

Step 3: Substitute and Simplify

$\cos 15^\circ = \left(\frac{\sqrt{2}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2}\right) \left(\frac{1}{2}\right)$ $= \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4}$ $= \frac{\sqrt{6} + \sqrt{2}}{4}$

Conclusion

The exact value of $\cos 15^\circ$ is: $\frac{\sqrt{6} - \sqrt{2}}{4}$

The answer you selected in the image is incorrect. The correct answer is $\frac{\sqrt{6} + \sqrt{2}}{4}$.

Would you like further details on this problem, or do you have any questions?

Here are some related questions to explore:

1. How can you find the exact value of $\cos 75^\circ$ using a sum or difference formula?
2. What is the exact value of $\sin 15^\circ$ using a similar approach?
3. Can you derive the value of $\tan 15^\circ$ using sum or difference formulas?
4. How would you calculate $\cos(30^\circ + 45^\circ)$ and does it lead to the same result?
5. What are other ways to compute trigonometric values for non-standard angles like $15^\circ$ and $75^\circ$?

Tip: Memorizing the values of common angles like $30^\circ$, $45^\circ$, and $60^\circ$ can make it easier to use sum and difference formulas for angles like $15^\circ$ and $75^\circ$.