# \frac{1-\cos(x)}{\sin(x)} is the formula of?

Shirley Thompson 2021-12-14 Answered
$\frac{1-\mathrm{cos}\left(x\right)}{\mathrm{sin}\left(x\right)}$ is the formula of?
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Daniel Cormack
$\left(1-\mathrm{cos}\left(x\right)\right)=2{\mathrm{sin}}^{2}\left(\frac{x}{2}\right)$
$\mathrm{sin}\left(x\right)=2\mathrm{sin}\left(\frac{x}{2}\right)\left(\mathrm{cos}\left(\frac{x}{2}\right)\right)$
Thus, we have:
$\frac{1-\mathrm{cos}\left(x\right)}{\mathrm{sin}\left(x\right)}=\frac{2{\mathrm{sin}}^{2}\left(\frac{x}{2}\right)}{2\mathrm{sin}\left(\frac{x}{2}\right)\left(\mathrm{cos}\left(\frac{x}{2}\right)\right)}=\mathrm{tan}\left(\frac{x}{2}\right)$
###### Not exactly what you’re looking for?
Melinda McCombs
$\frac{1-\mathrm{cos}\left(x\right)}{\mathrm{sin}\left(x\right)}$
$=\left(1-\mathrm{cos}\left(x\right)\right)\mathrm{csc}\left(x\right)$
Expand
$=\mathrm{csc}\left(x\right)-\mathrm{cos}\left(x\right)\mathrm{csc}\left(x\right)$
$=-\mathrm{cot}\left(x\right)+\mathrm{csc}\left(x\right)$