Pamela Meyer

2022-01-03

How can I prove the identity
$\frac{1}{1+\mathrm{sin}\left(x\right)}\equiv \frac{{\mathrm{sec}}^{2}\left(\frac{x}{2}\right)}{{\left(\mathrm{tan}\left(\frac{x}{2}\right)+1\right)}^{2}}$

Marcus Herman

Use
$\frac{\mathrm{sec}\frac{x}{2}}{\mathrm{tan}\frac{x}{2}+1}\equiv \frac{1}{\mathrm{sin}\frac{x}{2}+\mathrm{cos}\frac{x}{2}}$
${\left(\mathrm{sin}\frac{x}{2}+\mathrm{cos}\frac{x}{2}\right)}^{2}=?$

stomachdm

We can express any value trigonometric ratio of a given angle $2\theta$ in terms of the tangent of half that angle. For example:
$\mathrm{sin}2\theta \equiv \frac{2\mathrm{tan}\theta }{1+{\mathrm{tan}}^{2}\theta }$
$\mathrm{cos}2\theta \equiv \frac{1-{\mathrm{tan}}^{2}\theta }{1+{\mathrm{tan}}^{2}\theta }$
$\mathrm{tan}2\theta \equiv \frac{2\mathrm{tan}\theta }{1-{\mathrm{tan}}^{2}\theta }$
As an aside, this is useful in integration. So, you may want to use the first identity given to prove your identity.

Vasquez

Alternately, use half angle theorems:
$\mathrm{sin}x\equiv \frac{2\mathrm{tan}\left(\frac{x}{2}\right)}{1+{\mathrm{tan}}^{2}\left(\frac{x}{2}\right)}$

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