$f (x) = t a n^{- 1} (s i n x + c o s x)$ is an increasing function in

(0, \frac{π}{4})

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(0, \frac{π}{2})

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(\frac{- π}{4}, \frac{π}{4})

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None of these

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Solution

The correct options are
B $(0, \frac{π}{4})$
D $(\frac{- π}{4}, \frac{π}{4})$
$f (x) = t a n^{- 1} (s i n x + c o s x)$
$f^{1} (x) = \frac{1}{1 + (s i n x + c o s x)^{2}}$ $(c o s x - s i n x)$
always positive g(x)
$g (x) = c o s x - s i n x$
$g (x) = 0 \Rightarrow t a n x = 1$ $∴ x = \frac{π}{4}$
$∴ [\frac{- 3 π}{4}, \frac{π}{4}]$ increasing function
So, answers are $(0, \frac{π}{4}), (\frac{- π}{4}, \frac{π}{4})$

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