Answer:
\[\sin \left\{ 2{{\cot }^{-1}}\left( -\frac{5}{12} \right) \right\}=\sin \left\{ 2\left( \pi -{{\cot }^{-1}}\frac{5}{12} \right) \right\}\] \[=\sin \left\{ 2\pi -2{{\cot }^{-1}}\frac{5}{12} \right\}\] \[=-\sin \left\{ 2{{\cot }^{-1}}\frac{5}{12} \right\}\] \[=-\sin \left( 2ta{{n}^{-1}}\frac{12}{5} \right)\] \[[\because \,\,{{\cot }^{-1}}x={{\tan }^{-1}}\frac{1}{x},\,\,x>0]\] \[=-\sin \left[ \pi -{{\sin }^{-1}}\left( \frac{2\times \frac{12}{5}}{1+{{\left( \frac{12}{5} \right)}^{2}}} \right) \right]\] \[\left[ \because \,\,2{{\tan }^{-1}}x=\pi -{{\sin }^{-1}}\left( \frac{2x}{1+{{x}^{2}}} \right),x>1 \right]\] \[=-\sin \left( \pi -{{\sin }^{-1}}\frac{120}{169} \right)\] \[=-\sin \,\,{{\sin }^{-1}}\frac{120}{169}=\frac{-\,120}{169}\]
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