Solve the equation: `cos^(-1)(a/x)-cos^(-1)(b/x)=cos^(-1)(1/b)-cos^(-1)(1/a)`

Explanation

`cos^-1 (a/x) + cos^-1 (1/a) = cos^-1(b/x) + cos^-1 (1/b) `

`cos^-1 x + cos^-1 y = cos^-1 (xy - sqrt(1-x^2) sqrt(1-y^2))`

`cos^-1 (a/x 1/a - sqrt(1- a^2/x^2)sqrt(1- 1/a^2))`

`cos^-1 (b/x*1/b - sqrt(1- b^2/x^2) sqrt(1- 1/b^2))`

`(1- a^2/x^2) (1-1/a^2) = (1- b^2/x^2)(1- 1/b^2) `

`= 1- 1/a^2 - a^2/x^2 + 1/x^2 = 1- 1/b^2 - b^2/x^2 + 1/x^2`

`a^2/x^2 - b^2/x^2 = 1/b^2 - 1/a^2`

`= 1/x^2 [ a^2- b^2] = (a^2- b^2)/(a^2b^2) `

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