Simplyify `(sin75^(@)-sin15^(@))/(cos75^(@)+cos15^(@))`

Answer

To simplify the expression \((\sin 75^\circ - \sin 15^\circ) / (\cos 75^\circ + \cos 15^\circ)\), we can use the sine and cosine addition formulas.

### Step-by-Step Solution:

1. **Apply the Sine Difference Formula:**
\[ \sin A - \sin B = 2 \cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) \]
Here, let \(A = 75^\circ\) and \(B = 15^\circ\).
\[ \sin 75^\circ - \sin 15^\circ = 2 \cos\left(\frac{75^\circ + 15^\circ}{2}\right) \sin\left(\frac{75^\circ - 15^\circ}{2}\right) \]
\[ = 2 \cos(45^\circ) \sin(30^\circ) \]

2. **Calculate the Values:**
- \(\cos(45^\circ) = \frac{1}{\sqrt{2}}\)
- \(\sin(30^\circ) = \frac{1}{2}\)
\[ = 2 \cdot \frac{1}{\sqrt{2}} \cdot \frac{1}{2} = \frac{1}{\sqrt{2}} \]

3. **Apply the Cosine Sum Formula:**
\[ \cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) \]
\[ \cos 75^\circ + \cos 15^\circ = 2 \cos\left(\frac{75^\circ + 15^\circ}{2}\right) \cos\left(\frac{75^\circ - 15^\circ}{2}\right) \]
\[ = 2 \cos(45^\circ) \cos(30^\circ) \]

4. **Calculate the Values:**
- \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\)
\[ = 2 \cdot \frac{1}{\sqrt{2}} \cdot \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{\sqrt{2}} \]

5. **Combine the Results:**
Now we can substitute back into our original expression:
\[ \frac{\sin 75^\circ - \sin 15^\circ}{\cos 75^\circ + \cos 15^\circ} = \frac{\frac{1}{\sqrt{2}}}{\frac{\sqrt{3}}{\sqrt{2}}} \]
\[ = \frac{1}{\sqrt{3}} \]

6. **Final Simplification:**
\[ = \tan(30^\circ) = \frac{1}{\sqrt{3}} \]

Thus, the simplified value of the expression \((\sin 75^\circ - \sin 15^\circ) / (\cos 75^\circ + \cos 15^\circ)\) is \(\frac{1}{\sqrt{3}}\).