$\cot(\frac{A}{2}+\frac{B}{2})=\frac{\cot\frac{A}{2}\cot\frac{B}{2}-1}{\cot\frac{A}{2}+\cot\frac{B}{2}}$$\Rightarrow\frac{1}{\cot\frac{C}{2}}=\frac{\cot\frac{A}{2}\cot\frac{B}{2}-1}{\cot\frac{A}{2}+\cot\frac{B}{2}}$$\Rightarrow (1)$
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