# Solve the equation {tan x+cot y}{tan xcot x}=tan y+cot x

Solve the equation $\mathrm{tan}x+\mathrm{cot}y\mathrm{tan}x\mathrm{cot}x=\mathrm{tan}y+\mathrm{cot}x$
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faldduE
Work on the left side.
Rewrite as a sum:
$\frac{\mathrm{tan}x+\mathrm{cot}y}{\mathrm{tan}x\mathrm{cot}x}=\frac{\mathrm{tan}x}{\mathrm{tan}x}\mathrm{cot}y+\frac{\mathrm{cot}x}{\mathrm{tan}}x\mathrm{cot}y$
Simplify each term:
$\mathrm{tan}x+\mathrm{cot}y\mathrm{tan}x\mathrm{cot}y=\frac{1}{\mathrm{cot}y}+\frac{1}{\mathrm{tan}x}$
Use the reciprocal identities:
$\frac{\mathrm{tan}x+\mathrm{cot}y}{\mathrm{tan}x\mathrm{cot}x}=tany+\mathrm{cot}x$