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Question
If `sin^-1 (2a)/(1+a^2)+sin^-1 (2b)/(1+b^2)=2tan^-1x,` Prove that `x=(a+b)/(1-ab).`
Solution
Let: a = tan z
b = tan y
Then,
`sin^-1 (2a)/(1+a^2)+sin^-1 (2b)/(1+b^2)=2tan^-1x`
`=>sin^-1 (2tanz)/(1+tan^2z)+sin^-1 (2tany)/(1+tan^2y)=2tan^-1x`
`=>sin^-1(sin2z)+sin^-1(sin2y)=2tan^-1x` `[becausesin2x=(2tanx)/(1+tan^2x)]`
`=>2z+2y=2tan^-1x`
`=>tan^-1a+tan^-1b=tan^-1x` `[becausea=tanzandb=tany]`
`=>tan^-1 (a+b)/(1-ab)=tan^-1x` `[becausetan^-1x+tan^-1y=tan^-1 (x+y)/(1-xy)]`
`=>x=(a+b)/(1-ab)`
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