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