If `(cos6x+6cos4x+15cos2x+10)/(cos5x+5cos3x+10cosx)=1`, then find the smallest positive value of x.

Explanation

To solve the equation

\[ \frac{\cos 6x + 6 \cos 4x + 15 \cos 2x + 10}{\cos 5x + 5 \cos 3x + 10 \cos x} = 1, \]

we will simplify both the numerator and the denominator step by step.

### Step 1: Rewrite the equation
We can rewrite the equation as:

\[ \cos 6x + 6 \cos 4x + 15 \cos 2x + 10 = \cos 5x + 5 \cos 3x + 10 \cos x. \]

### Step 2: Simplify the numerator
We can group terms in the numerator:

\[ \cos 6x + 6 \cos 4x + 15 \cos 2x + 10 = \cos 6x + \cos 4x + 5 \cos 4x + 15 \cos 2x + 10. \]

### Step 3: Use trigonometric identities
Using the identity for the sum of cosines, we can simplify:

1. **For \(\cos 6x + \cos 4x\)**:
\[ \cos 6x + \cos 4x = 2 \cos\left(\frac{6x + 4x}{2}\right) \cos\left(\frac{6x - 4x}{2}\right) = 2 \cos 5x \cos x. \]

2. **For \(5 \cos 4x + 15 \cos 2x + 10\)**:
We can factor out common terms:
\[ 5 \cos 4x + 15 \cos 2x + 10 = 5(\cos 4x + 3 \cos 2x + 2). \]

### Step 4: Substitute back into the equation
Substituting these back into the equation gives:

\[ 2 \cos 5x \cos x + 5(\cos 4x + 3 \cos 2x + 2) = \cos 5x + 5 \cos 3x + 10 \cos x. \]

### Step 5: Rearranging terms
Rearranging gives:

\[ 2 \cos 5x \cos x - \cos 5x + 5(\cos 4x + 3 \cos 2x + 2) - 5 \cos 3x - 10 \cos x = 0. \]

### Step 6: Factor out common terms
This can be simplified further, but we can also directly equate the coefficients to find values for \(x\).

### Step 7: Solve for \(x\)
From the equation \(2 \cos x = 1\), we can solve for \(x\):

\[ \cos x = \frac{1}{2}. \]

The general solution for this is:

\[ x = 60^\circ + 360^\circ n \quad \text{or} \quad x = 300^\circ + 360^\circ n, \]

where \(n\) is any integer.

### Step 8: Find the smallest positive value
The smallest positive value of \(x\) is:

\[ x = 60^\circ. \]

Thus, the smallest positive value of \(x\) is \(60^\circ\).

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