Question

Prove that f(x) = tan-1(sin x+ cos x) is an increasing function in (-π/2,π/4)

Answer 1

Question :-

$\small \: prove \: that \: f(x) = { \tan}^{ - 1} ( sin \: x+ cos \: x ) \: \\ is \: an \: increasing \: function \: in \: \\ \bigg( \frac{ - \pi}{2} \: \frac{ \pi}{4} \bigg)$

Solution :-

We have,

$f(x) = { \tan}^{ - 1} \big( \: sin \: x \: + cos \: x \big)$

Therefore,

$\: \small {f}^{'} (x) = \frac{1}{1 + {(sin \: x \: + cos \: x)}^{2} } (cos \: x \: - sin \: x) \\ \\ \small \: {f}^{'} (x) = \frac{cos \: x - sin \: x}{2 + sin \: 2x}$

if f'(x) > 0 then f(x) is increasing function,

for ,

$\small \implies \: \frac{ - \pi}{2} < x < \frac{ \pi}{4} ,cos x > sin x$

Hence,

$\bf{f \: (x) \: if \: increasing \: in \: ( \frac{ - \pi}{2} , \frac{ \pi}{4} ) }$