If \( \cos ^{-1} \frac{1}{x}=\theta \), then \( \tan \theta= \)

1.	If \( \cos ^{-1} \frac{1}{x}=\theta \), then \( \tan \theta= \)
Answer» cos^-1\(\frac 1x\) = \(\theta\) ⇒ \(\frac 1x\) = cos \(\theta\) ⇒ sec \(\theta\) = x ⇒ sec²\(\theta\) - 1 = x² - 1 ⇒ tan²\(\theta\) = x² - 1 (\(\because\) tan²\(\theta\) = sec²\(\theta\) - 1) ⇒ tan \(\theta\) = \(\sqrt{x^2-1}\)

Answer»

cos^-1\(\frac 1x\) = \(\theta\)

⇒ \(\frac 1x\) = cos \(\theta\)

⇒ sec \(\theta\) = x

⇒ sec²\(\theta\) - 1 = x² - 1

⇒ tan²\(\theta\) = x² - 1 (\(\because\) tan²\(\theta\) = sec²\(\theta\) - 1)

⇒ tan \(\theta\) = \(\sqrt{x^2-1}\)

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If \( \cos ^{-1} \frac{1}{x}=\theta \), then \( \tan \theta= \)

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