Certainly! Let's use the sine difference identity to calculate \(\sin 15^\circ + \cos 15^\circ\).

### Step-by-Step Calculation Using the Sine and Cosine Difference Formulas

#### 1. Using the Difference Formula for Sine

Recall that \(15^\circ = 45^\circ - 30^\circ\). The sine of a difference identity is given by:

\[
\sin(a - b) = \sin a \cos b - \cos a \sin b
\]

For \(a = 45^\circ\) and \(b = 30^\circ\):

\[
\sin 15^\circ = \sin(45^\circ - 30^\circ) = \sin 45^\circ \cos 30^\circ - \cos 45^\circ \sin 30^\circ
\]

Now, substitute the known values of these trigonometric functions:

\[
\sin 45^\circ = \frac{\sqrt{2}}{2}, \quad \cos 45^\circ = \frac{\sqrt{2}}{2}
\]
\[
\cos 30^\circ = \frac{\sqrt{3}}{2}, \quad \sin 30^\circ = \frac{1}{2}
\]

Substitute these values into the sine difference formula:

\[
\sin 15^\circ = \left(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}\right) - \left(\frac{\sqrt{2}}{2} \cdot \frac{1}{2}\right)
\]

Simplify the expression:

\[
\sin 15^\circ = \frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4} = \frac{\sqrt{6} - \sqrt{2}}{4}
\]

#### 2. Using the Difference Formula for Cosine

Next, we use the cosine of a difference identity:

\[
\cos(a - b) = \cos a \cos b + \sin a \sin b
\]

For \(a = 45^\circ\) and \(b = 30^\circ\):

\[
\cos 15^\circ = \cos(45^\circ - 30^\circ) = \cos 45^\circ \cos 30^\circ + \sin 45^\circ \sin 30^\circ
\]

Substitute the same known values:

\[
\cos 15^\circ = \left(\frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2}\right) + \left(\frac{\sqrt{2}}{2} \cdot \frac{1}{2}\right)
\]

Simplify the expression:

\[
\cos 15^\circ = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4}
\]

#### 3. Add \(\sin 15^\circ\) and \(\cos 15^\circ\)

Now, let's add \(\sin 15^\circ\) and \(\cos 15^\circ\):

\[
\sin 15^\circ + \cos 15^\circ = \frac{\sqrt{6} - \sqrt{2}}{4} + \frac{\sqrt{6} + \sqrt{2}}{4}
\]

Combine the fractions:

\[
\sin 15^\circ + \cos 15^\circ = \frac{(\sqrt{6} - \sqrt{2}) + (\sqrt{6} + \sqrt{2})}{4}
\]

Simplify by combining like terms:

\[
\sin 15^\circ + \cos 15^\circ = \frac{2\sqrt{6}}{4} = \frac{\sqrt{6}}{2}
\]

### Final Answer

\[
\sin 15^\circ + \cos 15^\circ = \frac{\sqrt{6}}{2}
\]

### Conclusion

We used the sine and cosine difference formulas to find that:

\[
\sin 15^\circ + \cos 15^\circ = \frac{\sqrt{6}}{2}
\]

Would you like further details or have any questions?

### Follow-Up Questions

1. What is the value of \(\sin 75^\circ + \cos 75^\circ\)?
2. How would you use the sine and cosine sum identities for a different angle, like \(45^\circ + 30^\circ\)?
3. Can you prove the result using the Pythagorean identity?
4. How can this result be used to solve other trigonometric equations?
5. What happens when you add \(\sin x + \cos x\) for other specific angles like \(30^\circ\) or \(60^\circ\)?

### Tip

Knowing different trigonometric identities, like angle sum and difference formulas, can make solving complex trigonometric problems much easier!

Learn how to calculate sin 15 degrees + cos 15 degrees using the sine and cosine difference identities step-by-step. This problem involves trigonometric concepts including the sine and cosine of a difference identity, suitable for high school students.

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