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Question
Write the following function in the simplest form:
`tan^(-1) (sqrt(1 + x^2) - 1)/x, x ≠ 0`
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Solution
`tan^(-1) (sqrt(1 + x^2) - 1)/x`
Put x = tan θ ⇒ θ = tan–1 x
∴ `tan^(-1) (sqrt(1 + x^2) - 1)/x`
= `tan^(-1) ((sqrt(1 + tan^2 θ) - 1)/tan θ)`
= `tan^(-1) ((sec θ - 1)/tan θ)`
= `tan^(-1) ((1 - cos θ)/sin θ)`
= `tan^(-1) ((2 sin^2 θ/2)/(2 sin θ/2 cos θ/2))`
= `tan^(-1) (tan θ/2)`
= `θ/2`
= `1/2 tan^(-1) x`
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