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Question
Prove that
`sin{tan^-1 (1-x^2)/(2x)+cos^-1 (1-x^2)/(2x)}=1`
Solution
`sin{tan^-1 ((1-x^2)/(2x))+cos^-1 ((1-x^2)/(1+x^2))}=1`
LHS = `sin{tan^-1 ((1-x^2)/(2x))+cos^-1 ((1-x^2)/(1+x^2))}`
`=sin{sin^-1(((1-x^2)/(2x))/sqrt(1+(1-x^2)/(2x)))+cos^-1((1-x^2)/(1+x^2))}` `[becausetan^-1x=sin^-1 x/sqrt(1+x^2)]`
`=sin{sin^-1((1-x^2)/(1+x))+cos^1((1-x^2)/(1+x^2))}`
`=sin{pi/2}` `[becausesin^-1x+cos^-1x=pi/2]`
= 1 = RHS
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