Question Number 37244 by subhendubera201314@gmail.com last updated on 11/Jun/18
$$\mathrm{sin}^{\mathrm{2}} \left(\pi/\mathrm{11}\right)+\mathrm{sin}^{\mathrm{2}} \left(\mathrm{2}\pi/\mathrm{11}\right)+…+\mathrm{sin}^{\mathrm{2}} \left(\mathrm{5}\pi/\mathrm{11}\right)=? \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 11/Jun/18
$${sin}^{\mathrm{2}} {a}+{sin}^{\mathrm{2}} \mathrm{2}{a}+{sin}^{\mathrm{2}} \mathrm{3}{a}+{sin}^{\mathrm{2}} \mathrm{4}{a}+{sin}^{\mathrm{2}} \mathrm{5}{a} \\ $$$${a}=\frac{\Pi}{\mathrm{11}} \\ $$$$=\frac{\mathrm{1}−{cos}\mathrm{2}{a}}{\mathrm{2}}+\frac{\mathrm{1}−{cos}\mathrm{4}{a}}{\mathrm{2}}+\frac{\mathrm{1}−{cos}\mathrm{6}{a}}{\mathrm{2}}+\frac{\mathrm{1}−{cos}\mathrm{8}{a}}{\mathrm{2}}+ \\ $$$$\frac{\mathrm{1}−{cos}\mathrm{10}{a}}{\mathrm{2}} \\ $$$$=\mathrm{5}×\frac{\mathrm{1}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{2}}\left({c}\mathrm{2}{a}+{c}\mathrm{4}{a}+{c}\mathrm{6}{a}+{c}\mathrm{8}{a}+{c}\mathrm{10}{a}\right) \\ $$$${when}\:{c}\mathrm{2}{a}={cos}\mathrm{2}{a}\:{etc} \\ $$$${s}={c}\mathrm{2}{a}+{c}\mathrm{4}{a}+{c}\mathrm{6}{a}+{c}\mathrm{8}{a}+{c}\mathrm{10}{a} \\ $$$${s}×\mathrm{2}{sina}=\mathrm{2}{sina}.\left({c}\mathrm{2}{a}+{c}\mathrm{4}{a}+{c}\mathrm{6}{a}+{c}\mathrm{8}{a}+{c}\mathrm{10}{a}\right) \\ $$$${s}×\mathrm{2}{sina}=\mathrm{2}{sina}.{cos}\mathrm{2}{a}+\mathrm{2}{sina}.{cos}\mathrm{4}{a}+ \\ $$$$\:\:\mathrm{2}{sinacos}\mathrm{6}{a}+\mathrm{2}{sina}.{cos}\mathrm{8}{a}+\mathrm{2}{sina}.{cos}\mathrm{10}{a} \\ $$$${now}\:{look} \\ $$$$\mathrm{2}{sina}.{cos}\mathrm{2}{a}={sin}\mathrm{3}{a}−{sina} \\ $$$$\mathrm{2}{sina}.{cos}\mathrm{4}{a}={sin}\mathrm{5}{a}−{sin}\mathrm{3}{a} \\ $$$$\mathrm{2}{sina}.{cos}\mathrm{6}{a}={sin}\mathrm{7}{a}−{sin}\mathrm{5}{a} \\ $$$$\mathrm{2}{sina}.{cos}\mathrm{8}{a}={sin}\mathrm{9}{a}−{sin}\mathrm{7}{a} \\ $$$$\mathrm{2}{sina}.{cos}\mathrm{10}{a}={sin}\mathrm{11}{a}−{sin}\mathrm{9}{a} \\ $$$$ \\ $$$${now}\:{add}\:{them}…{addition}\:{of}\:{right}\:{side}\:{is} \\ $$$$={sin}\mathrm{11}{a}−{sina} \\ $$$$=\mathrm{2}{cos}\mathrm{6}{a}.{sin}\mathrm{5}{a} \\ $$$$ \\ $$$${addition}\:{of}\:{left}\:{is}\:{s}×\mathrm{2}{sina} \\ $$$${so} \\ $$$${s}×\mathrm{2}{sina}=\mathrm{2}{cos}\mathrm{6}{a}.{sin}\mathrm{5}{a} \\ $$$${s}=\frac{{cos}\mathrm{6}{a}.{sin}\mathrm{5}{a}\:}{{sina}} \\ $$$$\frac{\mathrm{5}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{2}}\left(\frac{{cos}\mathrm{6}{a}.{sin}\mathrm{5}{a}}{{sina}}\right) \\ $$$$=\frac{\mathrm{5}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{4}}\left(\frac{\mathrm{2}{cos}\mathrm{6}{a}.{sin}\mathrm{5}{a}}{{sina}}\right) \\ $$$$=\frac{\mathrm{5}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{4}}\left(\frac{{sin}\mathrm{11}{a}−{sina}}{{sina}}\right) \\ $$$${a}=\frac{\Pi}{\mathrm{11}}\:\:{so}\:\mathrm{11}{a}=\Pi \\ $$$${sin}\mathrm{11}{a}={sin}\Pi=\mathrm{0} \\ $$$$=\frac{\mathrm{5}}{\mathrm{2}}−\frac{\mathrm{1}}{\mathrm{4}}\left(\frac{\mathrm{0}−{sina}}{{sina}}\right) \\ $$$$=\frac{\mathrm{5}}{\mathrm{2}}+\frac{\mathrm{1}}{\mathrm{4}} \\ $$$$=\frac{\mathrm{10}+\mathrm{1}}{\mathrm{4}}=\frac{\mathrm{11}}{\mathrm{4}} \\ $$$$ \\ $$