In a triangle \( A B C \), if \( r_{1}+r_{3}+r=r_{2} \), then find ...

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In a triangle \( A B C \), if \( r_{1}+r_{3}+r=r_{2} \), then find the value of \( \left(\sec ^{2} A+\csc ^{2} B-\cot ^{2} \mathrm{C}\right) \).
\( \mathrm{P} \) [Note:All symbols used have usual meanings in a triangle.]

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