<s>[INST] What should be the method for solving and solution of $\frac{x^2+y^2}{x^2-y^2}$? If $\frac{x+y}{x-y}=3$
then $\frac{x^2+y^2}{x^2-y^2} = ?$
 [/INST] $$\text{As }\frac{x+y}{x-y}=\frac31$$
Applying componendo and dividendo,  
$$\frac{x+y+x-y}{x+y-(x-y)}=\frac{3+1}{3-1}\implies \frac xy=2$$
$$\implies \frac{x^2}{y^2}=\frac41$$
Again, applying componendo and dividendo, $$\frac{x^2+y^2}{x^2-y^2} = \frac{4+1}{4-1}=\frac53$$
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