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Question
Prove the following:
`2tan^-1(1/3) = tan^-1(3/4)`
Solution
L.H.S. = `2tan^-1(1/3)`
`= tan^-1[(2(1/3))/(1 - (1/3)^2)] ...[∵ 2tan^-1 x = tan^-1((2x)/(1 - x^2))]`
= `tan^-1[((2/3))/(1 - 1/9)]`
= `tan^-1(2/3 xx 9/8)`
= `tan^-1(3/4)`
= R.H.S.
Alternative Method:
L.H.S. = `2tan^-1(1/3) = tan^-1(1/3) + tan^-1(1/3)`
= `tan^-1[(1/3 + 1/3)/(1 - 1/3 xx 1/3)]`
= `tan^-1((3 + 3)/(9 - 1))`
= `tan^-1(6/8)`
= `tan^-1(3/4)`
= R.H.S.
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