If y sin (sin x) and `(d^(2)y)/(dx^(2))+(dy)/(dx)` tan x + f(x) = 0, then find f(x).

Explanation

To solve the problem, we need to find the function \( f(x) \) given the equation:

\[ \frac{d^2y}{dx^2} + \frac{dy}{dx} \tan x + f(x) = 0 \]

where \( y = \sin(\sin x) \).

### Step 1: Differentiate \( y \)

First, we differentiate \( y \) with respect to \( x \):

\[ y = \sin(\sin x) \]

Using the chain rule, we find the first derivative \( \frac{dy}{dx} \):

\[ \frac{dy}{dx} = \cos(\sin x) \cdot \frac{d}{dx}(\sin x) = \cos(\sin x) \cdot \cos x \]

### Step 2: Differentiate \( \frac{dy}{dx} \)

Next, we differentiate \( \frac{dy}{dx} \) to find \( \frac{d^2y}{dx^2} \):

\[ \frac{d^2y}{dx^2} = \frac{d}{dx} \left( \cos(\sin x) \cdot \cos x \right) \]

Using the product rule:

\[ \frac{d^2y}{dx^2} = \frac{d}{dx}(\cos(\sin x)) \cdot \cos x + \cos(\sin x) \cdot \frac{d}{dx}(\cos x) \]

Now, differentiate \( \cos(\sin x) \) using the chain rule:

\[ \frac{d}{dx}(\cos(\sin x)) = -\sin(\sin x) \cdot \cos x \]

And differentiate \( \cos x \):

\[ \frac{d}{dx}(\cos x) = -\sin x \]

Putting it all together:

\[ \frac{d^2y}{dx^2} = (-\sin(\sin x) \cdot \cos x) \cdot \cos x + \cos(\sin x) \cdot (-\sin x) \]

This simplifies to:

\[ \frac{d^2y}{dx^2} = -\sin(\sin x) \cdot \cos^2 x - \sin x \cdot \cos(\sin x) \]

### Step 3: Substitute into the original equation

Now, we substitute \( \frac{d^2y}{dx^2} \) and \( \frac{dy}{dx} \) into the original equation:

\[ -\sin(\sin x) \cdot \cos^2 x - \sin x \cdot \cos(\sin x) + \left( \cos(\sin x) \cdot \cos x \right) \tan x + f(x) = 0 \]

### Step 4: Simplify the equation

We can express \( \tan x \) as \( \frac{\sin x}{\cos x} \):

\[ -\sin(\sin x) \cdot \cos^2 x - \sin x \cdot \cos(\sin x) + \cos(\sin x) \cdot \cos x \cdot \frac{\sin x}{\cos x} + f(x) = 0 \]

This simplifies to:

\[ -\sin(\sin x) \cdot \cos^2 x - \sin x \cdot \cos(\sin x) + \sin x \cdot \cos(\sin x) + f(x) = 0 \]

The terms \( -\sin x \cdot \cos(\sin x) \) and \( +\sin x \cdot \cos(\sin x) \) cancel each other out:

\[ -\sin(\sin x) \cdot \cos^2 x + f(x) = 0 \]

### Step 5: Solve for \( f(x) \)

Rearranging gives us:

\[ f(x) = \sin(\sin x) \cdot \cos^2 x \]

### Final Answer

Thus, the function \( f(x) \) is:

\[ f(x) = \sin(\sin x) \cdot \cos^2 x \]

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