Prove that `(sin (A - B))/( sin A sin B ) + ( sin (B -C))/( sin B sin C ) + (sin (C - A))/( sin C sin A) =0`

Answer

To prove the equation

\[ \frac{\sin(A - B)}{\sin A \sin B} + \frac{\sin(B - C)}{\sin B \sin C} + \frac{\sin(C - A)}{\sin C \sin A} = 0, \]

we will use the sine subtraction formula and manipulate the expression step by step.

### Step 1: Apply the Sine Subtraction Formula
We start by using the identity for sine of a difference:

\[ \sin(A - B) = \sin A \cos B - \cos A \sin B. \]

Using this identity, we rewrite each term in the left-hand side (LHS):

\[ \frac{\sin(A - B)}{\sin A \sin B} = \frac{\sin A \cos B - \cos A \sin B}{\sin A \sin B}. \]

Thus, we can express the LHS as:

\[ \frac{\sin A \cos B}{\sin A \sin B} - \frac{\cos A \sin B}{\sin A \sin B} + \frac{\sin(B - C)}{\sin B \sin C} + \frac{\sin(C - A)}{\sin C \sin A}. \]

### Step 2: Rewrite the Other Terms
Now, we apply the sine subtraction formula to the other two terms:

1. For \( \sin(B - C) \):

\[ \sin(B - C) = \sin B \cos C - \cos B \sin C, \]

which gives:

\[ \frac{\sin(B - C)}{\sin B \sin C} = \frac{\sin B \cos C - \cos B \sin C}{\sin B \sin C}. \]

2. For \( \sin(C - A) \):

\[ \sin(C - A) = \sin C \cos A - \cos C \sin A, \]

which gives:

\[ \frac{\sin(C - A)}{\sin C \sin A} = \frac{\sin C \cos A - \cos C \sin A}{\sin C \sin A}. \]

### Step 3: Combine All Terms
Now we can combine all the terms:

\[ \frac{\sin A \cos B}{\sin A \sin B} - \frac{\cos A \sin B}{\sin A \sin B} + \frac{\sin B \cos C}{\sin B \sin C} - \frac{\cos B \sin C}{\sin B \sin C} + \frac{\sin C \cos A}{\sin C \sin A} - \frac{\cos C \sin A}{\sin C \sin A}. \]

### Step 4: Simplify Each Fraction
Now we can simplify each term:

1. The first term becomes \( \frac{\cos B}{\sin B} = \cot B \).
2. The second term becomes \( -\frac{\cos A}{\sin A} = -\cot A \).
3. The third term becomes \( \frac{\cos C}{\sin C} = \cot C \).
4. The fourth term becomes \( -\frac{\cos B}{\sin B} = -\cot B \).
5. The fifth term becomes \( \frac{\cos A}{\sin A} = \cot A \).
6. The sixth term becomes \( -\frac{\cos C}{\sin C} = -\cot C \).

### Step 5: Combine Like Terms
Now we can combine like terms:

\[ \cot A - \cot A + \cot B - \cot B + \cot C - \cot C = 0. \]

### Conclusion
Thus, we have shown that:

\[ \frac{\sin(A - B)}{\sin A \sin B} + \frac{\sin(B - C)}{\sin B \sin C} + \frac{\sin(C - A)}{\sin C \sin A} = 0. \]

Therefore, we conclude that the original statement is proved.