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# Differentiate Cos − 1 { Cos X + Sin X √ 2 } , − π 4 < X < π 4 ? - CBSE (Arts) Class 12 - Mathematics

ConceptSimple Problems on Applications of Derivatives

#### Question

Differentiate $\cos^{- 1} \left\{ \frac{\cos x + \sin x}{\sqrt{2}} \right\}, - \frac{\pi}{4} < x < \frac{\pi}{4}$ ?

#### Solution

$\text{ Let, y } = \cos^{- 1} \left\{ \frac{\cos x + \sin x}{\sqrt{2}} \right\}$

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

$y = \cos^{- 1} \left\{ \cos\frac{\pi}{4}\cos x + \sin\frac{\pi}{4}\sin x \right\}$

$y = \cos^{- 1} \left\{ \cos\left( \frac{\pi}{4} - x \right) \right\} . . . \left( i \right)$

$\text{ Here }, - \frac{\pi}{4} < x < \frac{\pi}{4}$

$\Rightarrow \frac{\pi}{4} > - x > - \frac{\pi}{4}$

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

$\Rightarrow \left( - \frac{\pi}{4} + \frac{\pi}{4} \right) < \left( - x + \frac{\pi}{4} \right) < \left( \frac{\pi}{4} + \frac{\pi}{4} \right)$

$\Rightarrow 0 < \left( \frac{\pi}{4} - x \right) < \frac{\pi}{2}$

$\text{ So, from equation } \left( i \right),$

$y = \frac{\pi}{4} - x \left[ \text{ Since }, \cos^{- 1} \left( \cos\theta \right) = \theta, \text{ if }\theta \in \left[ 0, \pi \right] \right]$

$\text{ Differentiating it with respect to x },$

$\frac{d y}{d x} = 0 - 1$

$\frac{d y}{d x} = - 1$

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Solution Differentiate Cos − 1 { Cos X + Sin X √ 2 } , − π 4 < X < π 4 ? Concept: Simple Problems on Applications of Derivatives.
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