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If y = sin (sin x), prove that : \(\frac{d^2y}{dx^2}+tanx.\frac{dy}{dx}+ycos^2x=0\)

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Given,

y = sin (sin x) ……equation 1

To prove:\(\frac{d^2y}{dx^2}+tanx.\frac{dy}{dx}+ycos^2x=0\) 

We notice a second-order derivative in the expression to be proved so first take the step to find the second order derivative.

Let’s find \(\frac{d^2y}{dx^2}\)

 As, \(\frac{d^2y}{dx^2}=\frac{d}{dx}(\frac{dy}{dx})\)

So, lets first find dy/dx

\(\frac{dy}{dx}=\frac{d}{dx}sin(sinx)\) 

Using chain rule, we will differentiate the above expression

Let t = sin x⟹\(\frac{dt}{dx}=cosx\)

Again differentiating with respect to x applying product rule:

Using chain rule again in the next step-

[using equation 1 : y =sin (sin x)]

And using equation 2, we have:

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