Question Number 204618 by es last updated on 23/Feb/24
$${if}\:\:\mathrm{7}{x}=\frac{\pi}{\mathrm{2}}\rightarrow\frac{{cosxsin}\mathrm{2}{xtan}\mathrm{3}{x}}{{cot}\mathrm{4}{xcos}\mathrm{5}{xsin}\mathrm{6}{x}}=? \\ $$
Answered by A5T last updated on 23/Feb/24
![(((cos(x)sin(2x)sin(3x))/(cos(3x)))/((cos(4x)cos(5x)sin(6x))/(sin(4x)))) =((sin(4x)cos(x)sin(2x)sin(3x))/(cos(3x)cos(4x)cos(5x)sin(6x)))=1 [since sin(4x)=cos(3x);cos(x)=sin(6x); sin(2x)=cos(5x);sin(3x)=cos(4x)]](https://www.tinkutara.com/question/Q204620.png)
$$\frac{\frac{{cos}\left({x}\right){sin}\left(\mathrm{2}{x}\right){sin}\left(\mathrm{3}{x}\right)}{{cos}\left(\mathrm{3}{x}\right)}}{\frac{{cos}\left(\mathrm{4}{x}\right){cos}\left(\mathrm{5}{x}\right){sin}\left(\mathrm{6}{x}\right)}{{sin}\left(\mathrm{4}{x}\right)}} \\ $$$$=\frac{{sin}\left(\mathrm{4}{x}\right){cos}\left({x}\right){sin}\left(\mathrm{2}{x}\right){sin}\left(\mathrm{3}{x}\right)}{{cos}\left(\mathrm{3}{x}\right){cos}\left(\mathrm{4}{x}\right){cos}\left(\mathrm{5}{x}\right){sin}\left(\mathrm{6}{x}\right)}=\mathrm{1} \\ $$$$\left[{since}\:{sin}\left(\mathrm{4}{x}\right)={cos}\left(\mathrm{3}{x}\right);{cos}\left({x}\right)={sin}\left(\mathrm{6}{x}\right);\right. \\ $$$$\left.{sin}\left(\mathrm{2}{x}\right)={cos}\left(\mathrm{5}{x}\right);{sin}\left(\mathrm{3}{x}\right)={cos}\left(\mathrm{4}{x}\right)\right] \\ $$