Key Concepts and Formulas

• Trigonometric Identity: $\sin(a+b) = \sin a \cos b + \cos a \sin b$
• Integral of cotangent: $\int \cot x \, dx = \log |\sin x| + C$

Step-by-Step Solution

Step 1: Rewrite the integrand using a clever trick. We want to manipulate the given integral to a form we can easily integrate. We add and subtract $\alpha$ from the argument of the sine function in the numerator. This allows us to use the sine addition formula. $\int \frac{\sin x}{\sin(x-\alpha)} dx = \int \frac{\sin(x - \alpha + \alpha)}{\sin(x - \alpha)} dx$

Step 2: Apply the sine addition formula. Now, we use the trigonometric identity $\sin(a+b) = \sin a \cos b + \cos a \sin b$, where $a = x - \alpha$ and $b = \alpha$. $\int \frac{\sin(x - \alpha)\cos \alpha + \cos(x - \alpha)\sin \alpha}{\sin(x - \alpha)} dx$

Step 3: Separate the fraction into two terms. We divide the numerator by the denominator, separating the integral into two simpler integrals. $\int \left( \frac{\sin(x - \alpha)\cos \alpha}{\sin(x - \alpha)} + \frac{\cos(x - \alpha)\sin \alpha}{\sin(x - \alpha)} \right) dx$ $= \int \left( \cos \alpha + \sin \alpha \cot(x - \alpha) \right) dx$

Step 4: Integrate each term. We integrate each term with respect to $x$. Remember that $\cos \alpha$ and $\sin \alpha$ are constants. $\int \cos \alpha \, dx + \int \sin \alpha \cot(x - \alpha) \, dx$ $= \cos \alpha \int dx + \sin \alpha \int \cot(x - \alpha) \, dx$ $= (\cos \alpha) x + (\sin \alpha) \log |\sin(x - \alpha)| + C$

Step 5: Compare with the given form. We are given that $\int \frac{\sin x}{\sin(x - \alpha)} dx = Ax + B \log |\sin(x - \alpha)| + C$ Comparing this with our result, $(\cos \alpha) x + (\sin \alpha) \log |\sin(x - \alpha)| + C$ we can identify $A = \cos \alpha$ and $B = \sin \alpha$. Therefore, $(A, B) = (\cos \alpha, \sin \alpha)$.

Common Mistakes & Tips

• Trigonometric Identities: Be careful with applying trigonometric identities, especially the sine and cosine addition/subtraction formulas. A mistake here can throw off the entire solution.
• Integration Constants: Don't forget the constant of integration, C. Although it doesn't affect the values of A and B in this problem, it's crucial for indefinite integrals.
• Careful with Signs: Pay close attention to the signs in the trigonometric identities and the integration formulas.

Summary

We started by rewriting the integrand using the sine addition formula. This allowed us to separate the integral into two parts, one involving a constant and the other involving the cotangent function. Integrating each part and comparing the result with the given form, we found the values of A and B to be $\cos \alpha$ and $\sin \alpha$, respectively.

Final Answer The final answer is \boxed{(\cos \alpha, \sin \alpha)}, which corresponds to option (B).