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: