Find y2 - 1-y2 and y4 - 1-y4 if y - 1-y - 9

y+1y=9 Squaring both side (y+1y)^2=9^2 y^2+1y^2+2× y×1y=81 y^2+1y^2=81 2=79 Squaring (y^2+1y^2)^2=(79)^2 y^4+1y^4+2× y^2×1y^2=6241 ...

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(a + b)^2 Expansion and Visualisation

Find y2 + 1...

Question

Find $y^{2} + \frac{1}{y^{2}}$ and $y^{4} + \frac{1}{y^{4}}$ if $y + \frac{1}{y} = 9$

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\sqrt{1 - x^{4}} + \sqrt{1 - y^{4}} = k (x^{2} - y^{2}),

\frac{d y}{d x} = \frac{x \sqrt{1 - y^{4}}}{y \sqrt{1 - x^{4}}}

1 + \frac{1}{y} + \frac{1}{y^{2}} + \frac{1}{y^{3}} + \frac{1}{y^{4}} + \frac{1}{y^{5}}

x^{2} + y^{2} = t + \frac{1}{t}

x^{4} + y^{4} = t^{2} + \frac{1}{t^{2}}

\frac{d y}{d x} = - \frac{y}{x}

x = \sqrt{a} + \frac{1}{\sqrt{a}}, y = \sqrt{a} - \frac{1}{\sqrt{a}}

x^{4} + y^{4} - 2 x^{2} y^{2}

{lim}_{y \to 1} (\frac{1}{y^{2} - 1} - \frac{2}{y^{4} - 1}) =

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