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provide solution for RD Sharma maths class 12 chapter Differentiation exercise  10.2 question 68

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Hint: you must know the rules of derivative of logarithm and trigonometric functions.

Given:   y=\frac{1}{2} \log \left(\frac{1-\cos 2 x}{1+\cos 2 x}\right)

Prove: \frac{d y}{d x}=2 \operatorname{cosec} 2 x


y=\frac{1}{2} \log \left(\frac{1-\cos 2 x}{1+\cos 2 x}\right)                    \left[\therefore 1-\cos 2 x=2 \sin ^{2} x ; 1+\cos 2 x=2 \cos ^{2} x\right]

\begin{aligned} &y=\frac{1}{2} \log \left(\frac{2 \sin ^{2} x}{2 \cos ^{2} x}\right) \\\\ &y=\frac{1}{2} \log \tan ^{2} x \end{aligned}

\begin{aligned} &y=\frac{1}{2} \times 2 \log \tan x \\\\ &y=\log \tan x \end{aligned}


Differentiate with respect to x,use chain rule

\begin{aligned} &\frac{d y}{d x}=\frac{d}{d x}(\log \tan x) \\\\ &\frac{d y}{d x}=\frac{1}{\tan x} \frac{d}{d x}(\tan x) \end{aligned}

\begin{aligned} &\frac{d y}{d x}=\frac{1}{\tan x} \times \sec ^{2} x \\\\ &\frac{d y}{d x}=\frac{\cos x}{\sin x} \times \frac{1}{\cos ^{2} x} \\\\ &\frac{d y}{d x}=\frac{1}{\sin x \cos x} \end{aligned}


Multiply and divide by 2

\frac{d y}{d x}=\frac{2}{2 \sin x \cos x}                            [\therefore 2 \sin x \cos x=\sin 2 x]

\frac{d y}{d x}=\frac{2}{\sin 2 x}                \left[\therefore \frac{1}{\sin x}=\operatorname{cosec} x\right]

\frac{d y}{d x}=2 \operatorname{cosec} 2 x

∴ Proved

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