SOLUTION: 2x= {{{ sqrt (a) }}} + 1/{{{ sqrt (a) }}} Prove {{{ sqrt (x^2 - 1)/ (x - sqrt(x^2-1))}}} = (a -1)/2 or (1-a)/2a

Algebra ->  Trigonometry-basics -> SOLUTION: 2x= {{{ sqrt (a) }}} + 1/{{{ sqrt (a) }}} Prove {{{ sqrt (x^2 - 1)/ (x - sqrt(x^2-1))}}} = (a -1)/2 or (1-a)/2a       Log On


   

Question 472205: 2x= +sqrt+%28a%29+ + 1/+sqrt+%28a%29+
Prove +sqrt+%28x%5E2+-+1%29%2F+%28x+-+sqrt%28x%5E2-1%29%29 = (a -1)/2 or (1-a)/2a
Answer by robertb(5830) About Me  (Show Source):
You can put this solution on YOUR website!
Consider the quadratic equation with roots sqrt%28a%29 and 1%2Fsqrt%28a%29.
The equation is given by y%5E2+-+2xy+%2B+1+=+0.
Using the quadraticformula, the roots of the preceding equation are
x-sqrt%28x%5E2+-+1%29 and x%2Bsqrt%28x%5E2+-+1%29 .
The second root is equal to 1%2F%28x-sqrt%28x%5E2+-+1%29%29, after rationalizing the numerator .
Hence we can let sqrt%28a%29+=+x-sqrt%28x%5E2+-+1%29 and 1%2Fsqrt%28a%29+=+1%2F%28x-sqrt%28x%5E2+-+1%29%29, without loss of generality.
Hence