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!