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Question
Show that f(x) = tan x is an increasing function on (−π/2, π/2) ?
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Solution
\[f\left( x \right) = \tan x\]
\[f'\left( x \right) = \sec^2 x\]
\[\text { Here },\]
\[\frac{- \pi}{2} < x < \frac{\pi}{2}\]
\[ \Rightarrow \sec x > 0 \left[ \because Sec \text { function is positive in first and fourth quadrant } \right]\]
\[ \Rightarrow \sec^2 x > 0\]
\[ \Rightarrow f'\left( x \right) > 0, \forall x \in \left( \frac{- \pi}{2}, \frac{\pi}{2} \right)\]
\[\text { So },f(x)\text { is increasing on } \left( \frac{- \pi}{2}, \frac{\pi}{2} \right) .\]
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