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Let's solve each expression step by step.

### (a) \( \cos(105^\circ) \cdot \sin(75^\circ) \)

We can break down the angles using known identities or exact trigonometric values:

- \( \cos(105^\circ) = \cos(180^\circ - 75^\circ) = -\sin(75^\circ) \)
  (This uses the identity \( \cos(180^\circ - x) = -\sin(x) \).)

Now we have:
\[
\cos(105^\circ) \cdot \sin(75^\circ) = (-\sin(75^\circ)) \cdot \sin(75^\circ) = -\sin^2(75^\circ)
\]

Using the exact value \( \sin(75^\circ) = \sin(90^\circ - 15^\circ) = \cos(15^\circ) \), we know:
\[
\cos(15^\circ) = \frac{\sqrt{6} + \sqrt{2}}{4}
\]

So:
\[
\sin^2(75^\circ) = \left(\frac{\sqrt{6} + \sqrt{2}}{4}\right)^2 = \frac{8 + 2\sqrt{12}}{16} = \frac{4 + \sqrt{3}}{4}
\]

Thus, the value of \( \cos(105^\circ) \cdot \sin(75^\circ) \) becomes:
\[
-\frac{4 + \sqrt{3}}{4}
\]

### (b) \( \sin(75^\circ) + \sin(15^\circ) \)

We can use the sum-to-product identity here:
\[
\sin(A) + \sin(B) = 2 \sin\left(\frac{A+B}{2}\right) \cdot \cos\left(\frac{A-B}{2}\right)
\]
For \( A = 75^\circ \) and \( B = 15^\circ \), we get:
\[
\sin(75^\circ) + \sin(15^\circ) = 2 \sin\left(\frac{75^\circ + 15^\circ}{2}\right) \cdot \cos\left(\frac{75^\circ - 15^\circ}{2}\right)
\]
\[
= 2 \sin(45^\circ) \cdot \cos(30^\circ)
\]
We know:
- \( \sin(45^\circ) = \frac{\sqrt{2}}{2} \)
- \( \cos(30^\circ) = \frac{\sqrt{3}}{2} \)

Thus:
\[
\sin(75^\circ) + \sin(15^\circ) = 2 \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{2}
\]

### Final Answers:
- (a) \( \cos(105^\circ) \cdot \sin(75^\circ) = -\frac{4 + \sqrt{3}}{4} \)
- (b) \( \sin(75^\circ) + \sin(15^\circ) = \frac{\sqrt{6}}{2} \)

Would you like more details or have any questions?

Here are 5 related questions for practice:
1. What is the exact value of \( \sin(105^\circ) \)?
2. Simplify \( \cos(75^\circ) \cdot \cos(15^\circ) \).
3. Use a sum or difference identity to find \( \sin(90^\circ) \).
4. Calculate \( \cos(120^\circ) \cdot \sin(60^\circ) \).
5. Derive the value of \( \sin(45^\circ) + \cos(45^\circ) \).

**Tip:** Always remember to check if sum or difference identities can simplify trigonometric expressions before directly calculating values.

Find the exact value of each expression: (a) cos(105°) · sin(75°), (b) sin(75°) + sin(15°).

Learn how to calculate the exact value of cos(105°) · sin(75°) and sin(75°) + sin(15°) using trigonometric identities and exact values. Detailed solution provided for high school trigonometry students.

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