Answer the second question.

Answer the second question. X x X X

Consider LHS \frac{x^{2} - y^{2}}{x^{2} + y^{2}} + \frac{x^{2} + y^{2}}{x^{2} - y^{2}} = \sqrt{2} \Rightarrow \frac{{(x^{2} - y^{2})}^{2} + {(x^{2} + y^{2})}^{2}}{x^{4} - y^{4}} = \sqrt{2} \Rightarrow \frac{x^{4} + y^{4} - 2 x^{2} y^{2} + x^{4} + y^{4} + 2 x^{2} y^{2}}{x^{4} - y^{4}} = \sqrt{2} \Rightarrow \frac{2 x^{4} + 2 y^{4}}{x^{4} - y^{4}} = \sqrt{2} \Rightarrow \frac{x^{4} + y^{4}}{x^{4} - y^{4}} = \frac{\sqrt{2}}{2} . . . (1) \Rightarrow \frac{x^{4} - y^{4}}{x^{4} + y^{4}} = \frac{2}{\sqrt{2}} . . . (2) Adding (1) and (2), we get \Rightarrow \frac{x^{4} + y^{4}}{x^{4} - y^{4}} + \frac{x^{4} - y^{4}}{x^{4} + y^{4}} = \frac{\sqrt{2}}{2} + \frac{2}{\sqrt{2}} \Rightarrow \frac{{(x^{4} + y^{4})}^{2} + {(x^{4} - y^{4})}^{2}}{x^{8} - y^{8}} = \frac{2 + 4}{2 \sqrt{2}} \Rightarrow \frac{x^{8} + y^{8} + 2 x^{4} y^{4} + x^{8} + y^{8} - 2 x^{4} y^{4}}{x^{8} - y^{8}} = \frac{3}{\sqrt{2}} \Rightarrow \frac{2 x^{8} + 2 y^{8}}{x^{8} - y^{8}} = \frac{3}{\sqrt{2}} \Rightarrow \frac{x^{8} + y^{8}}{x^{8} - y^{8}} = \frac{3}{2 \sqrt{2}} \Rightarrow \frac{x^{8} - y^{8}}{x^{8} + y^{8}} = \frac{2 \sqrt{2}}{3} Note : a^{2} - b^{2} = (a + b) (a - b)