Which of the following holds goods for any triangle ABC ?

Options

A. `(cos A)/(a) + (cos B)/(b) + (cos C)/(c ) = (a^(2) + b^(2) + c^(2))/(2abc)`
B. `(sin A)/(a) + (sin B)/(b) + (sin C)/(c ) = (3)/(2R)`
C. `(cos A)/(a) + (cos B)/(b) + (cos C)/(c ) = (2abc)/(a^(2) + b^(2) + c^(2))``
D. `(sin 2A)/(a^(2)) = (sin 2B)/(b^(2)) = (sin 2C)/(c^(2))`

Answer

To determine which of the given options holds true for any triangle ABC, we will analyze the sine rule and the cosine rule.

### Step-by-Step Solution:

1. **Understanding the Sine Rule**:
The sine rule states that for any triangle ABC, the ratios of the lengths of the sides to the sines of their opposite angles are equal:
\[ \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} = 2R \]
where \( R \) is the circumradius of the triangle.

2. **Expressing the Sine Rule**:
From the sine rule, we can express:
\[ \frac{\sin A}{a} = \frac{1}{2R}, \quad \frac{\sin B}{b} = \frac{1}{2R}, \quad \frac{\sin C}{c} = \frac{1}{2R} \]
Thus, we can conclude:
\[ \frac{\sin A}{a} + \frac{\sin B}{b} + \frac{\sin C}{c} = \frac{1}{2R} + \frac{1}{2R} + \frac{1}{2R} = \frac{3}{2R} \]

3. **Understanding the Cosine Rule**:
The cosine rule states that for any triangle ABC:
\[ a^2 = b^2 + c^2 - 2bc \cos A \]
\[ b^2 = a^2 + c^2 - 2ac \cos B \]
\[ c^2 = a^2 + b^2 - 2ab \cos C \]

4. **Expressing the Cosine Rule**:
We can rearrange the cosine rule to express \( \cos A \), \( \cos B \), and \( \cos C \):
\[ \cos A = \frac{b^2 + c^2 - a^2}{2bc}, \quad \cos B = \frac{a^2 + c^2 - b^2}{2ac}, \quad \cos C = \frac{a^2 + b^2 - c^2}{2ab} \]

5. **Analyzing the Options**:
- For the sine rule, we derived that \( \frac{\sin A}{a} + \frac{\sin B}{b} + \frac{\sin C}{c} = \frac{3}{2R} \) holds true for any triangle.
- For the cosine rule, the expressions for \( \cos A \), \( \cos B \), and \( \cos C \) do not yield a simple sum or ratio that holds universally for all triangles.

6. **Conclusion**:
The sine rule holds true for any triangle ABC, while the cosine rule does not yield a similar universal statement. Therefore, the correct options are those related to the sine rule.

### Final Answer:
The options that hold true for any triangle ABC are based on the sine rule.