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Prove that tan^(-1)((1)/(sqrt(x^(2) -1))...

Prove that `tan^(-1)((1)/(sqrt(x^(2) -1))) = (pi)/(2) - sec^(-1) x, x gt 1`

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`tan^(-1).(1)/(sqrt(x^(2) -1))`
Let `x = sec theta. " Then " theta = sec^(-1) x`
We have `tan^(-1).(1)/(sqrt(x^(2) -1)) = tan^(-1).(1)/(sqrt(sec^(2) theta -1))`
`= tan^(-1).(1)/(sqrt(sec^(2) theta -1))`
`= tan^(-1).(1)/(sqrt(tan^(2) theta))`
`= tan^(-1).(1)/(sqrt(tan^(2) theta))`
`= tan^(-1). (1)/(tan theta) " " ( :' x gt 1)`
`= tan^(-1) cot theta`
`= tan^(-1) tan ((pi)/(2) - theta)`
`= (pi)/(2) - theta = (pi)/(2) - sec^(-1) x`
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CENGAGE PUBLICATION-INVERSE TRIGONOMETRIC FUNCTIONS-All Questions
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