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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)/(4)xx(2)/(9))`
` tan^(-1). (9+8)/(36-2) = tan^(-1).(1)/(2) = (1)/(2) .2 tan ^(-1).(1)/(2)`
` = (1)/(2)cos^(-1).(1-(1)/(2^(2)))/(1+(1)/(2^(2))) (because cos ^(-1) .(1 -x^(2))/(1+ x^(2)) = 2 tan^(-1)x)`
` (1)/(2) cos^(-1).(3//4)/(5//4) = (1)/(2) cos^(-1).(3)/(5) = R.H.S`
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