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  • Provide Solution For R.D.Sharma Maths Class 12 Chapter 10 Differentiation Exercise 10.3 Question 21 Maths Textbook Solution.

Provide Solution For R.D.Sharma Maths Class 12 Chapter 10 Differentiation Exercise 10.3  Question 21 Maths Textbook Solution.

Answers (1)

Answer:\frac{dy}{dx}=\frac{1}{2}

Hint:

\frac{dy}{dx}\left ( x^{n} \right )=nx^{n-1}

\frac{dy}{dx}\left (constant\right )=0

Given:

\tan ^{-1}\left\{\frac{\sin x}{1+\cos x}\right\},-\pi<x<\pi

Solution:

Let,

y=\tan ^{-1}\left\{\frac{\sin x}{1+\cos x}\right\}

Function y is defined for all the real numbers where \cos x\neq -1

\begin{aligned} &\text { Using } 2 \cos ^{2} \theta=1+\cos 2 \theta \\ &2 \sin \theta \cos \theta=\sin 2 \theta \\ &y=\tan ^{-1}\left\{\frac{2 \sin \frac{x}{2} \cos \frac{x}{2}}{2 \cos ^{2} \frac{x}{2}}\right\} \\ &y=\tan ^{-1}\left\{-\frac{\sin x / 2}{\cos x / 2}\right\} \\ &y=\tan ^{-1}\left\{\tan \frac{x}{2}\right\} \\ &y=\frac{x}{2} \end{aligned}                                                

Differentiating with respect to x , We get

\frac{dy}{dx}=\frac{d}{dx}\left ( \frac{x}{2} \right )

\frac{dy}{dx}=\frac{1}{2}

 

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