let ((x^2 +y^2 )/(x^2 −y^2 ))+((x^2 −y^2 )/(x^2 +y^2 ))=k find the value of ((x^8 +y^8 )/(x^8 −y^8 ))+((x^8 −y^8 )/(x^8 +y^8 )) in terms of k


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Question Number 137563 by liberty last updated on 04/Apr/21

$l e t \frac{x^{2} + y^{2}}{x^{2} - y^{2}} + \frac{x^{2} - y^{2}}{x^{2} + y^{2}} = k$ $f i n d t h e v a l u e o f \frac{x^{8} + y^{8}}{x^{8} - y^{8}} + \frac{x^{8} - y^{8}}{x^{8} + y^{8}}$ $i n t e r m s o f k$
	Answered by Rasheed.Sindhi last updated on 04/Apr/21

	$l e t \frac{x^{2} + y^{2}}{x^{2} - y^{2}} + \frac{x^{2} - y^{2}}{x^{2} + y^{2}} = k$ $f i n d t h e v a l u e o f \frac{x^{8} + y^{8}}{x^{8} - y^{8}} + \frac{x^{8} - y^{8}}{x^{8} + y^{8}}$ $- - - - - - -$ $\frac{{(x^{2} + y^{2})}^{2} + {(x^{2} - y^{2})}^{2}}{(x^{2} + y^{2}) (x^{2} - y^{2})} = k$ $\frac{2 (x^{4} + y^{4})}{x^{4} - y^{4}} = k$ $\frac{x^{4} + y^{4}}{x^{4} - y^{4}} = \frac{k}{2}$ $\frac{x^{4} + y^{4}}{x^{4} - y^{4}} + \frac{x^{4} - y^{4}}{x^{4} + y^{4}} = \frac{k}{2} + \frac{2}{k}$ $\frac{{(x^{4} + y^{4})}^{2} + {(x^{4} - y^{4})}^{2}}{(x^{4} + y^{4}) (x^{4} - y^{4})} = \frac{k^{2} + 4}{2 k}$ $\frac{2 (x^{8} + y^{8})}{x^{8} - y^{8}} = \frac{k^{2} + 4}{2 k}$ $\frac{x^{8} + y^{8}}{x^{8} - y^{8}} = \frac{k^{2} + 4}{4 k}$ $\frac{x^{8} + y^{8}}{x^{8} - y^{8}} + \frac{x^{8} - y^{8}}{x^{8} + y^{8}} = \frac{k^{2} + 4}{4 k} + \frac{4 k}{k^{2} + 4}$ $= \frac{{(k^{2} + 4)}^{2} + {(4 k)}^{2}}{4 k (k^{2} + 4)}$ $= \frac{k^{4} + 24 k^{2} + 16}{4 k (k^{2} + 4)}$
	Answered by bramlexs22 last updated on 04/Apr/21

	$T h e e q u a l i t y \frac{x^{2} + y^{2}}{x^{2} - y^{2}} + \frac{x^{2} - y^{2}}{x^{2} + y^{2}} = k$ $i m p l i e s \frac{{(x^{2} + y^{2})}^{2} + {(x^{2} - y^{2})}^{2}}{x^{4} - y^{4}} = k$ $h e n c e \frac{x^{4} + y^{4}}{x^{4} - y^{4}} = \frac{k}{2} . a n d {(\frac{x}{y})}^{4} = \frac{k + 2}{k - 2}$ $T h e r e f o r e$ $\frac{x^{8} + y^{8}}{x^{8} - y^{8}} + \frac{x^{8} - y^{8}}{x^{8} + y^{8}} = \frac{2 (x^{16} + y^{16})}{x^{16} - y^{16}}$ $= 2 (\frac{{(\frac{k + 2}{k - 2})}^{4} + 1}{{(\frac{k - 2}{k + 2})}^{4} - 1}) = 2 (\frac{{(k + 2)}^{4} + {(k - 2)}^{4}}{{(k + 2)}^{4} - {(k - 2)}^{4}})$
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