kolutastmr

2022-07-16

Transforming the equation $\mathrm{cot}x-\mathrm{cos}x=0$ into the form $\mathrm{cos}x\left(1-\mathrm{sin}x\right)=0$

Asdrubali2r

Expert

$\mathrm{cot}\left(X\right)-\mathrm{cos}\left(X\right)=0\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}\frac{\mathrm{cos}X}{\mathrm{sin}X}-\mathrm{cos}X=0\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}\mathrm{cos}X\left(\frac{1}{\mathrm{sin}X}-1\right)=0$
$\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}\frac{\mathrm{cos}X\left(1-\mathrm{sin}X\right)}{\mathrm{sin}X}=0\phantom{\rule{thickmathspace}{0ex}}⟹\phantom{\rule{thickmathspace}{0ex}}\mathrm{cos}X\left(1-\mathrm{sin}X\right)=0\cdot \mathrm{sin}X=0$
This applies when $\mathrm{sin}X\ne 0$. If it is zero, then $\mathrm{cot}X=\mathrm{\infty }$ so the equation is not satisfied

Do you have a similar question?