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Question
Write the following function in the simplest form:
`tan^(-1) (sqrt(1+x^2) -1)/x, x != 0`
Solution
`tan^(-1) (sqrt(1+x^2) -1)/x`
Put ` x = tan theta =>theta = tan^(-1) x`
`:. tan^(-1) (sqrt(1+x^2) - 1)/x = tan^(-1) ((sqrt(1 + tan^2 theta) - 1)/ tan theta)`
`= tan^(-1) ((sec theta -1)/tan theta) = tan^(-1) ((1 - cos theta)/ sin theta)`
`= tan^(-1) ((2 sin^2 theta/2)/(2 sin theta/2 cos theta/2))`
`= tan^(-1) (tan theta/2) = theta/2 = 1/2 tan^(-1) x`
∴ `tan^(-1) (sqrt(1 + x^2 - 1)/(x)) = 1/2 tan^(-1)x`
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