Solve `cos theta+cos 7 theta+cos 3theta+cos 5 theta=0`,

Explanation

To solve the equation \( \cos \theta + \cos 7\theta + \cos 3\theta + \cos 5\theta = 0 \), we can follow these steps:

### Step 1: Group the terms
We can group the terms in pairs:
\[ (\cos \theta + \cos 7\theta) + (\cos 3\theta + \cos 5\theta) = 0 \]

### Step 2: Apply the cosine addition formula
Using the identity \( \cos A + \cos B = 2 \cos \left( \frac{A+B}{2} \right) \cos \left( \frac{A-B}{2} \right) \), we apply it to both groups:
1. For \( \cos \theta + \cos 7\theta \):
\[ \cos \theta + \cos 7\theta = 2 \cos \left( \frac{\theta + 7\theta}{2} \right) \cos \left( \frac{\theta - 7\theta}{2} \right) = 2 \cos (4\theta) \cos (3\theta) \]

2. For \( \cos 3\theta + \cos 5\theta \):
\[ \cos 3\theta + \cos 5\theta = 2 \cos \left( \frac{3\theta + 5\theta}{2} \right) \cos \left( \frac{3\theta - 5\theta}{2} \right) = 2 \cos (4\theta) \cos (1\theta) \]

### Step 3: Combine the results
Now substituting back into the equation:
\[ 2 \cos (4\theta) \cos (3\theta) + 2 \cos (4\theta) \cos (1\theta) = 0 \]
Factoring out \( 2 \cos (4\theta) \):
\[ 2 \cos (4\theta) (\cos (3\theta) + \cos (1\theta)) = 0 \]

### Step 4: Set each factor to zero
This gives us two equations to solve:
1. \( \cos (4\theta) = 0 \)
2. \( \cos (3\theta) + \cos (1\theta) = 0 \)

### Step 5: Solve \( \cos (4\theta) = 0 \)
The cosine function equals zero at odd multiples of \( \frac{\pi}{2} \):
\[ 4\theta = \frac{\pi}{2} + n\pi \quad (n \in \mathbb{Z}) \]
Thus,
\[ \theta = \frac{\pi}{8} + \frac{n\pi}{4} \]

### Step 6: Solve \( \cos (3\theta) + \cos (1\theta) = 0 \)
Using the identity again:
\[ \cos (3\theta) = -\cos (1\theta) \]
This can be rewritten as:
\[ \cos (3\theta) + \cos (1\theta) = 0 \implies 2 \cos \left( \frac{3\theta + 1\theta}{2} \right) \cos \left( \frac{3\theta - 1\theta}{2} \right) = 0 \]
This leads to:
\[ 2 \cos (2\theta) \cos (1\theta) = 0 \]
Thus, we have:
1. \( \cos (2\theta) = 0 \)
2. \( \cos (1\theta) = 0 \)

### Step 7: Solve \( \cos (2\theta) = 0 \)
\[ 2\theta = \frac{\pi}{2} + m\pi \quad (m \in \mathbb{Z}) \]
Thus,
\[ \theta = \frac{\pi}{4} + \frac{m\pi}{2} \]

### Step 8: Solve \( \cos (1\theta) = 0 \)
\[ \theta = \frac{\pi}{2} + k\pi \quad (k \in \mathbb{Z}) \]

### Final Solutions
Combining all solutions, we have:
\[ \theta = \frac{\pi}{8}, \quad \theta = \frac{\pi}{4}, \quad \theta = \frac{\pi}{2} + k\pi \]