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.