Solve `sin x("cos"x/4-2 sin x)+(1+"sin"x/4-2 cos x)cos x =0`.

Explanation

To solve the equation \( \sin x \left( \cos \frac{x}{4} - 2 \sin x \right) + \left( 1 + \sin \frac{x}{4} - 2 \cos x \right) \cos x = 0 \), we will follow these steps:

### Step 1: Rewrite the Equation
We start with the given equation:
\[ \sin x \left( \cos \frac{x}{4} - 2 \sin x \right) + \left( 1 + \sin \frac{x}{4} - 2 \cos x \right) \cos x = 0 \]

### Step 2: Expand the Terms
Distributing the terms:
\[ \sin x \cos \frac{x}{4} - 2 \sin^2 x + \cos x + \sin \frac{x}{4} \cos x - 2 \cos^2 x = 0 \]

### Step 3: Group Similar Terms
Rearranging the equation:
\[ \sin x \cos \frac{x}{4} + \sin \frac{x}{4} \cos x - 2 \sin^2 x - 2 \cos^2 x + \cos x = 0 \]

### Step 4: Factor Out Common Terms
Notice that \( -2 \sin^2 x - 2 \cos^2 x \) can be factored:
\[ \sin x \cos \frac{x}{4} + \sin \frac{x}{4} \cos x - 2(\sin^2 x + \cos^2 x) + \cos x = 0 \]
Using the identity \( \sin^2 x + \cos^2 x = 1 \):
\[ \sin x \cos \frac{x}{4} + \sin \frac{x}{4} \cos x - 2 + \cos x = 0 \]

### Step 5: Simplify Further
Rearranging gives:
\[ \sin x \cos \frac{x}{4} + \sin \frac{x}{4} \cos x + \cos x - 2 = 0 \]

### Step 6: Use Trigonometric Identities
Using the identity for \( \sin(a + b) \):
\[ \sin x \cos \frac{x}{4} + \sin \frac{x}{4} \cos x = \sin \left( x + \frac{x}{4} \right) = \sin \left( \frac{5x}{4} \right) \]
Thus, we can rewrite the equation as:
\[ \sin \left( \frac{5x}{4} \right) + \cos x - 2 = 0 \]

### Step 7: Isolate the Sine Function
Rearranging gives:
\[ \sin \left( \frac{5x}{4} \right) + \cos x = 2 \]
Since the maximum value of \( \sin \) and \( \cos \) is 1, this equation has no solutions because \( \sin \left( \frac{5x}{4} \right) + \cos x \) cannot equal 2.

### Conclusion
The given equation has no solutions.