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# Solution for Show that F(X) = Sin X − Cos X is an Increasing Function on (−π/4, π/4) ? - CBSE (Commerce) Class 12 - Mathematics

ConceptIncreasing and Decreasing Functions

#### Question

Show that f(x) = sin x − cos x is an increasing function on (−π/4, π/4) ?

#### Solution

$f\left( x \right) = \sin x - \cos x$

$f'\left( x \right) = \cos x + \sin x$

$= \cos x\left( 1 + \frac{\sin x}{\cos x} \right)$

$= \cos x\left( 1 + \cot x \right)$

$\text { Here, }$

$\frac{- \pi}{4} < x < \frac{\pi}{4}$

$\Rightarrow \cos x > 0 . . . \left( 1 \right)$

$\text { Also, }$

$\frac{- \pi}{4} < x < \frac{\pi}{4} \Rightarrow - 1 < \cot x < 1$

$\Rightarrow 0 < 1 + \cot x < 2$

$\Rightarrow 1 + \cot x > 0 . . . \left( 2 \right)$

$\cos x\left( 1 + \cot x \right) > 0, \forall x \in \left( \frac{- \pi}{4}, \frac{\pi}{4} \right) \left[ \text { From eqs }. (1) \text { and }(2) \right]$

$\Rightarrow f'\left( x \right) > 0, \forall x \in \left( \frac{- \pi}{4}, \frac{\pi}{4} \right)$

$\text { So,}f\left( x \right) \text { is increasing on }\left( \frac{- \pi}{4}, \frac{\pi}{4} \right).$

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Solution Show that F(X) = Sin X − Cos X is an Increasing Function on (−π/4, π/4) ? Concept: Increasing and Decreasing Functions.
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