If $f(x)=\sin (\sin x)$ and $f^{\prime \prime}(x)+\tan x f^{\prime}(x)+g(x)$ $=0$, then $g(x)$ is :

If $f(x)=\sin (\sin x)$ and $f^{\prime \prime}(x)+\tan x f^{\prime}(x)+g(x)$ $=0$, then $g(x)$ is :
  1. $\cos ^2 x \cos (\sin x)$
  2. $\sin ^2 x \cos (\cos x)$
  3. $\sin ^2 x \sin (\cos x)$
  4. $\cos ^2 x \sin (\sin x)$

Solution

$ \begin{aligned} & \text { } f(x)=\sin (\sin x) \\ & \Rightarrow f^{\prime}(x)=\cos (\sin x) \cdot \cos x \\ & \Rightarrow f^{\prime \prime}(x)=-\sin (\sin x) \cdot \cos ^2 x+\cos (\sin x) \text {. } \\ & (-\sin x) \\ & =-\cos ^2 x \cdot \sin (\sin x)-\sin x \cdot \cos (\sin x) \\ & \text { Now } f^{\prime \prime}(x)+\tan x . f^{\prime}(x)+g(x)=0 \\ & \Rightarrow g(x)=\cos ^2 x \cdot \sin (\sin x)+\sin x \cdot \cos (\sin x) \\ & -\tan x \cdot \cos x \cdot \cos (\sin x) \\ & \Rightarrow g(x)=\cos ^2 x \cdot \sin (\sin x) \text {. } \\ & \end{aligned} $

Asked in: JEE Main 2013 (23 Apr Online)

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