# How do I prove \frac{\sin x+\csc y}{\sin y+\csc x}=\csc y

How do I prove $\frac{\mathrm{sin}x+\mathrm{csc}y}{\mathrm{sin}y+\mathrm{csc}x}=\mathrm{csc}y\mathrm{sin}x$?
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Alfred Mueller
If
$\frac{p+\frac{1}{q}}{q+\frac{1}{p}}=\frac{p}{q}$ if $pq+1\ne 0$
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votaren10
$\mathrm{csc}\left(y\right)\mathrm{sin}\left(x\right)=\mathrm{csc}\left(y\right)\mathrm{sin}\left(x\right)\cdot \frac{\mathrm{sin}\left(y\right)+\mathrm{csc}\left(x\right)}{\mathrm{sin}\left(y\right)+\mathrm{csc}\left(x\right)}$
$=\frac{\mathrm{csc}\left(y\right)\mathrm{sin}\left(x\right)\mathrm{sin}\left(y\right)+\mathrm{csc}\left(y\right)\mathrm{sin}\left(x\right)\mathrm{csc}\left(x\right)}{\mathrm{sin}\left(y\right)+\mathrm{csc}\left(x\right)}$
$=\frac{\mathrm{sin}\left(x\right)+\mathrm{csc}\left(y\right)}{\mathrm{sin}\left(y\right)+\mathrm{csc}\left(x\right)}$