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Let's solve the two given trigonometric expressions step by step.

### (a) \(\sin 105^\circ - \sin 15^\circ\)

We can use the **sum-to-product identities** for sine functions:
\[
\sin A - \sin B = 2 \cos \left( \frac{A + B}{2} \right) \sin \left( \frac{A - B}{2} \right)
\]

For \(A = 105^\circ\) and \(B = 15^\circ\):

\[
\sin 105^\circ - \sin 15^\circ = 2 \cos \left( \frac{105^\circ + 15^\circ}{2} \right) \sin \left( \frac{105^\circ - 15^\circ}{2} \right)
\]
\[
= 2 \cos \left( \frac{120^\circ}{2} \right) \sin \left( \frac{90^\circ}{2} \right)
\]
\[
= 2 \cos 60^\circ \sin 45^\circ
\]

Now, using the known values:

\[
\cos 60^\circ = \frac{1}{2}, \quad \sin 45^\circ = \frac{\sqrt{2}}{2}
\]

Substituting these:

\[
\sin 105^\circ - \sin 15^\circ = 2 \times \frac{1}{2} \times \frac{\sqrt{2}}{2} = \frac{\sqrt{2}}{2}
\]

So, the exact value of \(\sin 105^\circ - \sin 15^\circ\) is:
\[
\boxed{\frac{\sqrt{2}}{2}}
\]

### (b) \(\cos 52.5^\circ \cos 7.5^\circ\)

We use the **product-to-sum identities** for cosine functions:
\[
\cos A \cos B = \frac{1}{2} [\cos (A + B) + \cos (A - B)]
\]

For \(A = 52.5^\circ\) and \(B = 7.5^\circ\):

\[
\cos 52.5^\circ \cos 7.5^\circ = \frac{1}{2} \left[ \cos(52.5^\circ + 7.5^\circ) + \cos(52.5^\circ - 7.5^\circ) \right]
\]
\[
= \frac{1}{2} \left[ \cos 60^\circ + \cos 45^\circ \right]
\]

Now, using the known values:

\[
\cos 60^\circ = \frac{1}{2}, \quad \cos 45^\circ = \frac{\sqrt{2}}{2}
\]

Substituting these:

\[
\cos 52.5^\circ \cos 7.5^\circ = \frac{1}{2} \left( \frac{1}{2} + \frac{\sqrt{2}}{2} \right)
\]
\[
= \frac{1}{2} \times \frac{1 + \sqrt{2}}{2} = \frac{1 + \sqrt{2}}{4}
\]

So, the exact value of \(\cos 52.5^\circ \cos 7.5^\circ\) is:
\[
\boxed{\frac{1 + \sqrt{2}}{4}}
\]

---

Would you like any further details or explanations?

### 5 Related Questions:
1. How do you derive sum-to-product identities in trigonometry?
2. Can you explain other product-to-sum identities for different trigonometric functions?
3. What are the common trigonometric values for angles like 30°, 45°, and 60°?
4. How can you simplify trigonometric expressions using identities?
5. How can trigonometric identities help in solving real-life problems?

### Tip:
Always try to memorize common trigonometric values (e.g., \(\sin 30^\circ\), \(\cos 45^\circ\)) to speed up your calculations!

Find the exact value of sin(105°) - sin(15°) and cos(52.5°) * cos(7.5°).

Learn how to solve the trigonometric expressions sin(105°) - sin(15°) and cos(52.5°) * cos(7.5°) using sum-to-product and product-to-sum identities. This step-by-step solution is perfect for high school students studying trigonometry.

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