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Question
Prove that:
Solution
\[ \text{ LHS }= \frac{\sin\left( A - B \right)}{\sin A \sin B} + \frac{\sin\left( B - C \right)}{\sin B \sin C} + \frac{\sin\left( C - A \right)}{\sin C \sin A}\]
\[ = \frac{\sin A \cos B - \cos A \sin B}{\sin A \sin B} + \frac{\sin B \cos C - \cos B \sin C}{\sin B \sin C} + \frac{\sin C \cos A - \cos C \sin A}{\sin C \sin A}\]
\[ = \frac{\sin A \cos B}{\sin A \sin B} - \frac{\cos A \sin B}{\sin A \sin B} + \frac{\sin B \cos C}{\sin B \sin C} - \frac{\cos B \sin C}{\sin B \sin C} + \frac{\sin C \cos A}{\sin C \sin A} - \frac{\cos C \sin A}{\sin C \sin A}\]
\[ = \frac{\cos B}{\sin B} - \frac{\cos A}{\sin A} + \frac{\cos C}{\sin C} - \frac{\cos B}{\sin B} + \frac{\cos A}{\sin A} - \frac{\cos C}{\sin C}\]
\[ = cotB - cotA + cotC - cotB + cotA - cotC\]
\[ = 0\]
= RHS
Hence proved.
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