Question Number 42519 by gyugfeet last updated on 27/Aug/18
$${cosec}\mathrm{2}{A}+{cosec}\mathrm{4}{A}+{cosec}\mathrm{8}{A}={cotA}−{cot}\mathrm{8}{A}\left({prlve}\:{ghis}\right) \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 27/Aug/18
$${cotA}−\left({cosec}\mathrm{2}{A}+{cosec}\mathrm{4}{A}+{cosec}\mathrm{8}{A}\right) \\ $$$${cotA}−{cose}\mathrm{2}{A}−\left({cosec}\mathrm{4}{A}+{cosec}\mathrm{8}{A}\right) \\ $$$$\frac{{cosA}}{{sinA}}−\frac{\mathrm{1}}{\mathrm{2}{sinAcosA}}−\left({cosec}\mathrm{4}{A}+{cosec}\mathrm{8}{A}\right) \\ $$$$\frac{\mathrm{2}{cos}^{\mathrm{2}} {A}−\mathrm{1}}{\mathrm{2}{sinAcosA}}−\left({cosec}\mathrm{4}{A}+{cosec}\mathrm{8}{A}\right) \\ $$$$\frac{{cos}\mathrm{2}{A}}{{sin}\mathrm{2}{A}}−{cosec}\mathrm{4}{A}−{cosec}\mathrm{8}{A} \\ $$$${cot}\mathrm{2}{A}−{cosec}\mathrm{4}{A}−{cosec}\mathrm{8}{A} \\ $$$$\frac{{cos}\mathrm{2}{A}}{{sin}\mathrm{2}{A}}−\frac{\mathrm{1}}{{sin}\mathrm{4}{A}}−{cosec}\mathrm{8}{A} \\ $$$$\frac{\mathrm{2}{cos}^{\mathrm{2}} \mathrm{2}{A}−\mathrm{1}}{\mathrm{2}{sin}\mathrm{2}{Acos}\mathrm{2}{A}}−{cosec}\mathrm{8}{A} \\ $$$$ \\ $$$$\frac{{cos}\mathrm{4}{A}}{{sin}\mathrm{4}{A}}−\frac{\mathrm{1}}{{sin}\mathrm{8}{A}} \\ $$$$\frac{\mathrm{2}{cos}^{\mathrm{2}} \mathrm{4}{A}−\mathrm{1}}{{sin}\mathrm{8}{A}} \\ $$$$\frac{{cos}\mathrm{8}{A}}{{sin}\mathrm{8}{A}} \\ $$$${cot}\mathrm{8}{A}\:\:{proved} \\ $$$$ \\ $$