Question Number 189201 by Shrinava last updated on 13/Mar/23
$$\mathrm{In}\:\:\:\bigtriangleup\mathrm{ABC}\:\:\:\mathrm{holds}: \\ $$$$\sqrt{\mathrm{2}}\:\mathrm{a}\:\mathrm{cos}\:\frac{\mathrm{B}}{\mathrm{2}}\:\mathrm{cos}\:\frac{\mathrm{C}}{\mathrm{2}}\:=\:\mathrm{s} \\ $$$$\Rightarrow\:\mathrm{sec}\:\left(\mathrm{2B}\right)\:+\:\mathrm{tan}\:\left(\mathrm{2B}\right)\:=\:\frac{\mathrm{c}\:+\:\mathrm{b}}{\mathrm{c}\:−\:\mathrm{b}} \\ $$
Answered by som(math1967) last updated on 13/Mar/23
$$\sqrt{\mathrm{2}}{acos}\frac{{B}}{\mathrm{2}}{cos}\frac{{C}}{\mathrm{2}}={s} \\ $$$$\sqrt{\mathrm{2}}{a}\sqrt{\frac{{s}\left({s}−{b}\right)}{{ca}}}×\sqrt{\frac{{s}\left({s}−{c}\right)}{{ab}}}={s} \\ $$$$\sqrt{\mathrm{2}}{a}\frac{{s}}{{a}}×\sqrt{\frac{\left({s}−{b}\right)\left({s}−{c}\right)}{{bc}}}={s} \\ $$$$\sqrt{\mathrm{2}}\:{sin}\frac{{A}}{\mathrm{2}}=\mathrm{1} \\ $$$$\therefore{sin}\frac{{A}}{\mathrm{2}}=\frac{\mathrm{1}}{\:\sqrt{\mathrm{2}}}\Rightarrow\frac{{A}}{\mathrm{2}}=\frac{\pi}{\mathrm{4}} \\ $$$$\therefore{A}=\frac{\pi}{\mathrm{2}}\:\therefore{a}^{\mathrm{2}} ={b}^{\mathrm{2}} +{c}^{\mathrm{2}} \\ $$$${now}\:{sec}\mathrm{2}{B}+{tan}\mathrm{2}{B} \\ $$$$=\frac{\mathrm{1}}{{cos}\mathrm{2}{B}}+\frac{{sin}\mathrm{2}{B}}{{cos}\mathrm{2}{B}} \\ $$$$=\frac{\mathrm{1}+{sin}\mathrm{2}{B}}{{cos}\mathrm{2}{B}} \\ $$$$=\frac{\left({cosB}+{sinB}\right)^{\mathrm{2}} }{{cos}^{\mathrm{2}} {B}−{sin}^{\mathrm{2}} {B}} \\ $$$$=\frac{{cosB}+{sinB}}{{cosB}−{sinB}} \\ $$$$=\frac{{cosB}+{sin}\left(\frac{\pi}{\mathrm{2}}−{C}\right)}{{cosB}−{sin}\left(\frac{\pi}{\mathrm{2}}−{C}\right)}\:\:\left[{A}=\frac{\pi}{\mathrm{2}}\:\therefore{B}+{C}=\frac{\pi}{\mathrm{2}}\right] \\ $$$$=\frac{{cosB}+{cosC}}{{cosB}−{cosC}} \\ $$$$=\frac{\frac{{c}^{\mathrm{2}} +{a}^{\mathrm{2}} −{b}^{\mathrm{2}} }{\mathrm{2}{ca}}\:+\frac{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} −{c}^{\mathrm{2}} }{\mathrm{2}{ab}}}{\frac{{c}^{\mathrm{2}} +{a}^{\mathrm{2}} −{b}^{\mathrm{2}} }{\mathrm{2}{ca}}\:−\frac{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} −{c}^{\mathrm{2}} }{\mathrm{2}{ab}}} \\ $$$$=\frac{\frac{\mathrm{2}{c}^{\mathrm{2}} }{\mathrm{2}{ca}}+\frac{\mathrm{2}{b}^{\mathrm{2}} }{\mathrm{2}{ab}}}{\frac{\mathrm{2}{c}^{\mathrm{2}} }{\mathrm{2}{ca}}−\frac{\mathrm{2}{b}^{\mathrm{2}} }{\mathrm{2}{ab}}}\:\:\:\left[{a}^{\mathrm{2}} ={b}^{\mathrm{2}} +{c}^{\mathrm{2}} \right] \\ $$$$=\frac{{c}+{b}}{{c}−{b}} \\ $$
Commented by Shrinava last updated on 13/Mar/23
$$\mathrm{thankyou}\:\mathrm{dearSer}\:\mathrm{cool} \\ $$