Solve `2 sin^(3) x=cos x`.

Explanation

To solve the equation \( 2 \sin^3 x = \cos x \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ 2 \sin^3 x = \cos x \]

### Step 2: Divide both sides by \( \sin^3 x \)
Assuming \( \sin x \neq 0 \) (since \( \sin x = 0 \) does not satisfy the equation), we can divide both sides by \( \sin^3 x \):
\[ 2 = \frac{\cos x}{\sin^3 x} \]
This can be rewritten as:
\[ 2 = \cot x \cdot \sec^2 x \]

### Step 3: Use the identity for \( \sec^2 x \)
Recall that \( \sec^2 x = 1 + \tan^2 x \). Therefore, we can express \( \cot x \cdot \sec^2 x \) as:
\[ \cot x \cdot (1 + \tan^2 x) = \cot x + \cot x \tan^2 x \]
Since \( \cot x = \frac{1}{\tan x} \), we can write:
\[ \cot x \tan^2 x = \frac{\tan^2 x}{\tan x} = \tan x \]
Thus, we have:
\[ 2 = \cot x + \tan x \]

### Step 4: Substitute \( \tan x \) in terms of \( \cot x \)
Let \( y = \cot x \). Then, \( \tan x = \frac{1}{y} \), and we rewrite the equation:
\[ 2 = y + \frac{1}{y} \]

### Step 5: Multiply through by \( y \)
To eliminate the fraction, multiply through by \( y \):
\[ 2y = y^2 + 1 \]
Rearranging gives us:
\[ y^2 - 2y + 1 = 0 \]

### Step 6: Factor the quadratic equation
This can be factored as:
\[ (y - 1)^2 = 0 \]
Thus, we find:
\[ y - 1 = 0 \quad \Rightarrow \quad y = 1 \]

### Step 7: Solve for \( x \)
Since \( y = \cot x \), we have:
\[ \cot x = 1 \]
This occurs at:
\[ x = \frac{\pi}{4} + n\pi, \quad n \in \mathbb{Z} \]

### Final Solution
The general solution to the equation \( 2 \sin^3 x = \cos x \) is:
\[ x = n\pi + \frac{\pi}{4}, \quad n \in \mathbb{Z} \]