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If x lt 0, then prove that cos^(-1) x = ...

If `x lt 0`, then prove that `cos^(-1) x = pi + tan^(-1). (sqrt(1 - x^(2)))/(x)`

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To prove that \( \cos^{-1} x = \pi + \tan^{-1} \left( \frac{\sqrt{1 - x^2}}{x} \right) \) for \( x < 0 \), we can follow these steps: ### Step 1: Assume \( x = \cos \theta \) Since \( x < 0 \), we can assume that \( x = \cos \theta \) where \( \theta \) is in the range \( \left( \frac{\pi}{2}, \pi \right) \). ### Step 2: Express \( \theta \) in terms of \( x \) From our assumption, we have: \[ ...
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