Given x = `(sqrt(a^2 + b^2) + sqrt(a^2 - b^2))/(sqrt(a^2 + b^2) + sqrt(a^2 - b^2))`
Use componendo and dividendo to prove that b^2 = (2a^2x)/(x^2 + 1)
Solution
x = `(sqrt(a^2 + b^2) + sqrt(a^2 - b^2))/(sqrt(a^2 + b^2) + sqrt(a^2 - b^2))`
by componendo and dividendo
`(x + 1)/(x - 1) = (sqrt(a^2 + b^2) + sqrt(a^2 - b^2) + sqrt(a^2 + b^2) - sqrt(a^2 - b^2))/(sqrt(a^2 + b^2) + sqrt(a^2 - b^2) - sqrt(a^2 + b^2) + sqrt(a^2 - b^2))`
`(x + 1)/(x - 1) = (2sqrt(a^2 + b^2))/(2sqrt(a^2 - b^2))`
Squaring both sides
`(x^2 + 2x + 1)/(x^2 - 2x + 1) = (a^2 + b^2)/(a^2 - b^2)`
By componendo and dividendo
`((x^2 + 2x + 1) + (x^2 - 2x + 1))/((x^2 + 2x + 1) - (x^2 - 2x + 1)) = ((a^2 + b^2) + (a^2 - b^2))/((a^2 + b^2) - (a^2 - b^2))`
`=> (2(x^2 + 1))/(4x) = (2a^2)/(2b^2)`
`=> (x^2 + 1)/(2x) = a^2/b^2`
`=> b^2 = (2a^2 x)/(x^2 + 1)`
