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$$\mathrm{In}\:\mathrm{acute}\:\bigtriangleup\mathrm{ABC}\:\mathrm{holds}: \\ $$$$\underset{\boldsymbol{\mathrm{cyc}}} {\prod}\:\left(\mathrm{tan}\:\mathrm{A}\right)^{\boldsymbol{\mathrm{a}}} \:\:\leqslant\:\:\underset{\boldsymbol{\mathrm{cyc}}} {\prod}\:\left(\mathrm{cot}\:\frac{\mathrm{A}}{\mathrm{2}}\right)^{\frac{\boldsymbol{\mathrm{b}}\:+\:\boldsymbol{\mathrm{c}}}{\mathrm{2}}} \\ $$
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$$\mathrm{using}\:\mathrm{maclaurin}'\mathrm{s}\:\mathrm{series}\:\mathrm{exapand} \\ $$$$\frac{{x}}{\mathrm{2}}\left(\frac{{e}^{{x}} +\mathrm{1}}{{e}^{{x}} −\mathrm{1}}\right)\:\mathrm{upto}\:{x}^{\mathrm{4}} \:\mathrm{term} \\ $$
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$$ \\ $$$$\:\:\:\:\:\mathrm{I}{f}\:\:,\:{f}\left({x}\right)=\:\mid\:{x}\:−\:{k}\lfloor{x}\rfloor\mid\:{be}\: \\ $$$$\:{one}\:−{to}−{one}\:{function}\:{on}\:\left(\mathrm{0},\mathrm{2}\right) \\ $$$$\:\:\: \\ $$$$\:\:\:\:{find}\:{the}\:{value}\:{of}\:\:\:\:\:''\:\:\:\:{k}\:\:\:''. \\ $$$$ \\ $$
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$$\:\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\sqrt[{\mathrm{sin}\:{x}}]{\mathrm{1}+\mathrm{2sin}\:{x}}−\sqrt[{\mathrm{tan}\:{x}}]{\mathrm{1}+\mathrm{tan}\:{x}}}{\mathrm{2}{x}^{\mathrm{3}} }=? \\ $$
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