If $\frac{x}{y}=\frac{a+2}{a-2}$, then find the value of $\frac{x^2-y^2}{x^2+y^2}$.

Given:
$$\frac{x}{y}=\frac{a+2}{a-2}$$

Squaring both sides:

$$\frac{x^2}{y^2}=\frac{(a+2{)}^2}{(a-2{)}^2}=\frac{a^2+4a+4}{a^2-4a+4}$$

Using Componendo and Dividendo rule, specifically to find $\frac{x^2-y^2}{x^2+y^2}$:

$$\frac{x^2-y^2}{x^2+y^2}=\frac{(a^2+4a+4)-(a^2-4a+4)}{(a^2+4a+4)+(a^2-4a+4)}$$

Simplify the numerator and the denominator:

$$\text{Numerator}=a^2+4a+4-a^2+4a-4=8a$$

$$\text{Denominator}=a^2+4a+4+a^2-4a+4=2a^2+8=2(a^2+4)$$

Thus:

$$\frac{x^2-y^2}{x^2+y^2}=\frac{8a}{2(a^2+4)}=\frac{4a}{a^2+4}$$