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# Solution for Show that F(X) = Tan−1 (Sin X + Cos X) is a Decreasing Function on the Interval (π/4, π/2) ? - CBSE (Science) Class 12 - Mathematics

ConceptIncreasing and Decreasing Functions

#### Question

Show that f(x) = tan−1 (sin x + cos x) is a decreasing function on the interval (π/4, π/2) ?

#### Solution

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

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

$= \frac{1}{1 + 1 + 2 \sin x \cos x}\left( \cos x - \sin x \right)$

$= \frac{\left( \cos x - \sin x \right)}{2 + \sin 2x}$

$\text { Here },$

$\frac{\pi}{4} < x < \frac{\pi}{2}$

$\Rightarrow \frac{\pi}{2} < 2x < \pi$

$\Rightarrow \sin 2x > 0$

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

$\text { Also,}$

$\frac{\pi}{4} < x < \frac{\pi}{2}$

$\cos x < \sin x$

$\Rightarrow \cos x - \sin x < 0 . . . \left( 2 \right)$

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

$\text { So },f\left( x \right)\text { is decreasing on }\left( \frac{\pi}{4}, \frac{\pi}{2} \right).$

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Solution Show that F(X) = Tan−1 (Sin X + Cos X) is a Decreasing Function on the Interval (π/4, π/2) ? Concept: Increasing and Decreasing Functions.
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