 # simplify (cosx/1+sinx)+(1+sinx/cosx) DofotheroU 2020-12-28 Answered

simplify $\left(\mathrm{cos}x/1+\mathrm{sin}x\right)+\left(1+\mathrm{sin}x/\mathrm{cos}x\right)$

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$\left(\mathrm{cos}x/1+\mathrm{sin}x\right)+\left(1+\mathrm{sin}x/\mathrm{cos}x\right)=\left(\mathrm{cos}x/1+\mathrm{sin}x\right)+\left(\left(\mathrm{cos}x+\mathrm{sin}x\right)/\mathrm{cos}x\right)$
$=\left({\mathrm{cos}}^{2}x+\left(1+\mathrm{sin}x\right)\left(\mathrm{cos}x+\mathrm{sin}x\right)\right)/\mathrm{cos}x\left(1+\mathrm{sin}x\right)$
$=\left({\mathrm{cos}}^{2}x+\mathrm{cos}x+\mathrm{sin}x+\mathrm{sin}x\mathrm{cos}x+{\mathrm{sin}}^{2}x\right)/\mathrm{cos}x\left(1+\mathrm{sin}x\right)$
$=\left(1+\mathrm{cos}x+\mathrm{sin}x+\mathrm{sin}x\mathrm{cos}x\right)/\mathrm{cos}x\left(1+\mathrm{sin}x\right)$
$=\left(1+\mathrm{cos}x+\mathrm{sin}x\left(1+\mathrm{cos}x\right)\right)/\mathrm{cos}x\left(1+\mathrm{sin}x\right)$
$=\left(1+\mathrm{cos}x\right)\left(1+\mathrm{sin}x\right)/\mathrm{cos}x\left(1+\mathrm{sin}x\right)$
$=\left(1+\mathrm{cos}x\right)/\mathrm{cos}x$
$=1+\mathrm{sec}x$