Question

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

Answer 1

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

⇒ `cos^-1  a/x+cos^-1  1/a=cos^-1  1/b+cos^-1  b/x`

⇒  `cos^-1 [a/x  xx1/a-sqrt(1-(a/x)^2)sqrt(1-(1/a)^2)]=cos^-1[b/x  xx1/b-sqrt(1-(b/x)^2)sqrt(1-(1/b)^2)]`     `[because cos^-1x+cos^-1y=cos^-1(xy-sqrt(1-x^2)sqrt(1-y^2))]`

⇒  `cos^-1[1/x-sqrt(1-a^2/x^2)xxsqrt(1-1/a^2)]=cos^-1[1/x-sqrt(1-b^2/x^2)xxsqrt(1-1/b^2)]`

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

⇒  `(a^2-b^2)/x^2=(a^2-b^2)/(a^2b^2)`

⇒  `x^2=a^2b^2`

⇒  `x=ab`