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Prove that: tan^-1(1/4)+tan^-1(2/9)=1/2 ...

Prove that: `tan^-1(1/4)+tan^-1(2/9)=1/2 cos^-1(3/5)`.

Text Solution

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L.H.S. `=tan^(-1)(1/4)+tan^(-1)(2/9)`
`tan^(-1)((1/4+2/9)/(1-1/4xx2/9))`
`tan^(-1)(17/34)`
`=tan^(-1)(1/2)`
`=1/2(2tan^(-1)(1/2))`
`=1/2cos^(-1)((1-1/4)/(1+1/4)) (:' 2tan^(-1)x=cos^(-1)((1-x^2)/(1+x^2))`
`=1/2cos^(-1)(3/5)`
= R.H.S.
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