Solve : `cos 6 theta + cos 4 theta + cos 2 theta + 1 = 0, 0^(@) lt theta lt 180^(@)`

Answer

To solve the equation \( \cos 6\theta + \cos 4\theta + \cos 2\theta + 1 = 0 \) for \( 0^\circ < \theta < 180^\circ \), we can follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ \cos 6\theta + \cos 4\theta + \cos 2\theta + 1 = 0 \]

### Step 2: Group terms and use cosine sum formulas
We can group the first two cosine terms:
\[ \cos 6\theta + \cos 4\theta = 2 \cos\left(\frac{6\theta + 4\theta}{2}\right) \cos\left(\frac{6\theta - 4\theta}{2}\right) \]
This simplifies to:
\[ 2 \cos(5\theta) \cos(\theta) \]
Thus, we can rewrite the equation as:
\[ 2 \cos(5\theta) \cos(\theta) + \cos 2\theta + 1 = 0 \]

### Step 3: Substitute \( \cos 2\theta \)
Using the identity \( \cos 2\theta = 2\cos^2\theta - 1 \), we substitute:
\[ 2 \cos(5\theta) \cos(\theta) + (2\cos^2\theta - 1) + 1 = 0 \]
This simplifies to:
\[ 2 \cos(5\theta) \cos(\theta) + 2\cos^2\theta = 0 \]

### Step 4: Factor the equation
We can factor out \( 2\cos\theta \):
\[ 2\cos\theta (\cos(5\theta) + \cos\theta) = 0 \]

### Step 5: Solve each factor
1. **From \( 2\cos\theta = 0 \)**:
\[ \cos\theta = 0 \implies \theta = 90^\circ \]

2. **From \( \cos(5\theta) + \cos\theta = 0 \)**:
\[ \cos(5\theta) = -\cos\theta \]
This implies:
\[ \cos(5\theta) + \cos\theta = 0 \implies 2\cos\left(\frac{5\theta + \theta}{2}\right) \cos\left(\frac{5\theta - \theta}{2}\right) = 0 \]
Simplifying gives:
\[ 2\cos(3\theta) \cos(2\theta) = 0 \]

### Step 6: Solve each cosine factor
1. **From \( \cos(3\theta) = 0 \)**:
\[ 3\theta = 90^\circ + n \cdot 180^\circ \implies \theta = 30^\circ + n \cdot 60^\circ \]
For \( n = 0 \): \( \theta = 30^\circ \)
For \( n = 1 \): \( \theta = 90^\circ \)
For \( n = 2 \): \( \theta = 150^\circ \)

2. **From \( \cos(2\theta) = 0 \)**:
\[ 2\theta = 90^\circ + n \cdot 180^\circ \implies \theta = 45^\circ + n \cdot 90^\circ \]
For \( n = 0 \): \( \theta = 45^\circ \)
For \( n = 1 \): \( \theta = 135^\circ \)

### Step 7: Collect all solutions
The solutions we found are:
- From \( \cos\theta = 0 \): \( \theta = 90^\circ \)
- From \( \cos(3\theta) = 0 \): \( \theta = 30^\circ, 90^\circ, 150^\circ \)
- From \( \cos(2\theta) = 0 \): \( \theta = 45^\circ, 135^\circ \)

### Final Solutions
Thus, the complete set of solutions in the interval \( 0^\circ < \theta < 180^\circ \) is:
\[ \theta = 30^\circ, 45^\circ, 90^\circ, 135^\circ, 150^\circ \]