If $\cot \theta =\frac{x}{y}$, then what is the value of $\frac{x\cos \theta -y\sin \theta }{x\cos \theta +y\sin \theta }$?

Given expression:
$$\frac{x\cos \theta -y\sin \theta }{x\cos \theta +y\sin \theta }$$

Divide every individual term in both the numerator and the denominator by $\sin \theta$:

$$\frac{x\left(\frac{\cos \theta }{\sin \theta }\right)-y\left(\frac{\sin \theta }{\sin \theta }\right)}{x\left(\frac{\cos \theta }{\sin \theta }\right)+y\left(\frac{\sin \theta }{\sin \theta }\right)}=\frac{x\cot \theta -y}{x\cot \theta +y}$$

Since we are given that $\cot \theta =\frac{x}{y}$, we can substitute it into our derived equation:

$$\frac{x\left(\frac{x}{y}\right)-y}{x\left(\frac{x}{y}\right)+y}=\frac{\frac{x^2}{y}-y}{\frac{x^2}{y}+y}=\frac{\frac{x^2-y^2}{y}}{\frac{x^2+y^2}{y}}=\frac{x^2-y^2}{x^2+y^2}$$