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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

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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 for question: 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. For the courses CBSE (Science), CBSE (Commerce), PUC Karnataka Science, CBSE (Arts)
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