The value of $\frac{1}{(9-4\sqrt{5})^2}+\frac{1}{(9+4\sqrt{5})^2}$ is :

322

$\frac{1}{(9-4\sqrt{5})^2}+\frac{1}{(9+4\sqrt{5})^2}$ =

We know that,

(a + b)² = (a² + b² + 2ab)

(a - b)² = (a² + b² - 2ab)

(a² - b²) = (a + b) (a - b)

So according to the question,

$\frac{1}{(9-4\sqrt{5})^2}+\frac{1}{(9+4\sqrt{5})^2}$

= [(9 + 4$\sqrt {5}$)² + (9 - 4$\sqrt {5}$)²]/[(9 - 4$\sqrt {5}$)² (9 + 4$\sqrt {5}$)²]

= (81 + 80 + 72$\sqrt {5}$ + 81 + 80 - 72$\sqrt {5}$)/(81 - 80)

= 322