Find the value of `(sin75^(@)+sin15^(@))/(cos75^(@)+cos15^(@))` :

Options

A. 1
B. 2
C. `(sqrt3)/(2)`
D. can't be determined

Answer

To solve the expression \((\sin 75^\circ + \sin 15^\circ) / (\cos 75^\circ + \cos 15^\circ)\), we can use the sum-to-product identities for sine and cosine.

### Step-by-Step Solution:

1. **Apply the Sum-to-Product Identity for Sine:**
The formula for the sum of sines is:
\[ \sin A + \sin B = 2 \sin\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \]
Here, let \(A = 75^\circ\) and \(B = 15^\circ\):
\[ \sin 75^\circ + \sin 15^\circ = 2 \sin\left(\frac{75^\circ + 15^\circ}{2}\right) \cos\left(\frac{75^\circ - 15^\circ}{2}\right) \]
Simplifying this gives:
\[ \sin 75^\circ + \sin 15^\circ = 2 \sin(45^\circ) \cos(30^\circ) \]

2. **Calculate \(\sin(45^\circ)\) and \(\cos(30^\circ)\):**
We know:
\[ \sin(45^\circ) = \frac{\sqrt{2}}{2}, \quad \cos(30^\circ) = \frac{\sqrt{3}}{2} \]
Therefore:
\[ \sin 75^\circ + \sin 15^\circ = 2 \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{2} \]

3. **Apply the Sum-to-Product Identity for Cosine:**
The formula for the sum of cosines is:
\[ \cos A + \cos B = 2 \cos\left(\frac{A + B}{2}\right) \cos\left(\frac{A - B}{2}\right) \]
Using the same values for \(A\) and \(B\):
\[ \cos 75^\circ + \cos 15^\circ = 2 \cos\left(\frac{75^\circ + 15^\circ}{2}\right) \cos\left(\frac{75^\circ - 15^\circ}{2}\right) \]
This simplifies to:
\[ \cos 75^\circ + \cos 15^\circ = 2 \cos(45^\circ) \cos(30^\circ) \]

4. **Calculate \(\cos(45^\circ)\) and \(\cos(30^\circ)\):**
We know:
\[ \cos(45^\circ) = \frac{\sqrt{2}}{2}, \quad \cos(30^\circ) = \frac{\sqrt{3}}{2} \]
Therefore:
\[ \cos 75^\circ + \cos 15^\circ = 2 \cdot \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{6}}{2} \]

5. **Combine the Results:**
Now we can substitute back into our original expression:
\[ \frac{\sin 75^\circ + \sin 15^\circ}{\cos 75^\circ + \cos 15^\circ} = \frac{\frac{\sqrt{6}}{2}}{\frac{\sqrt{6}}{2}} = 1 \]

### Final Answer:
Thus, the value of \(\frac{\sin 75^\circ + \sin 15^\circ}{\cos 75^\circ + \cos 15^\circ}\) is \(1\).