A general solution of `tan^(2) theta+ cos 2 theta=1` is `(n in Z)`

Options

A. `n pi-pi/4`
B. `2npi+pi/4`
C. `npi+pi/4`
D. `n pi`

Explanation

To solve the equation \( \tan^2 \theta + \cos 2\theta = 1 \), we can follow these steps:

### Step 1: Use the identity for \( \cos 2\theta \)
We know that:
\[ \cos 2\theta = 1 - 2\sin^2 \theta \]
However, we can also express it in terms of \( \tan \theta \):
\[ \cos 2\theta = \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} \]
Thus, we can rewrite the equation as:
\[ \tan^2 \theta + \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} = 1 \]

### Step 2: Combine the terms
To combine the terms, we will take the common denominator:
\[ \tan^2 \theta + \frac{1 - \tan^2 \theta}{1 + \tan^2 \theta} = \frac{\tan^2 \theta (1 + \tan^2 \theta) + (1 - \tan^2 \theta)}{1 + \tan^2 \theta} \]
This simplifies to:
\[ \frac{\tan^2 \theta + \tan^4 \theta + 1 - \tan^2 \theta}{1 + \tan^2 \theta} = \frac{\tan^4 \theta + 1}{1 + \tan^2 \theta} \]

### Step 3: Set the equation to zero
Now, we set the equation to zero:
\[ \frac{\tan^4 \theta + 1}{1 + \tan^2 \theta} = 1 \]
This leads to:
\[ \tan^4 \theta + 1 = 1 + \tan^2 \theta \]
Simplifying gives:
\[ \tan^4 \theta - \tan^2 \theta = 0 \]

### Step 4: Factor the equation
We can factor this equation:
\[ \tan^2 \theta (\tan^2 \theta - 1) = 0 \]
This gives us two cases to consider.

### Step 5: Solve the first case
1. **Case 1:** \( \tan^2 \theta = 0 \)
- This implies \( \tan \theta = 0 \).
- The general solution for this is:
\[ \theta = n\pi, \quad n \in \mathbb{Z} \]

### Step 6: Solve the second case
2. **Case 2:** \( \tan^2 \theta - 1 = 0 \)
- This implies \( \tan^2 \theta = 1 \) or \( \tan \theta = \pm 1 \).
- The general solutions for this are:
\[ \theta = n\pi + \frac{\pi}{4}, \quad n \in \mathbb{Z} \quad \text{(for } \tan \theta = 1\text{)} \]
\[ \theta = n\pi - \frac{\pi}{4}, \quad n \in \mathbb{Z} \quad \text{(for } \tan \theta = -1\text{)} \]

### Final General Solution
Combining both cases, the general solution of the equation \( \tan^2 \theta + \cos 2\theta = 1 \) is:
\[ \theta = n\pi \quad \text{or} \quad \theta = n\pi + \frac{\pi}{4} \quad \text{or} \quad \theta = n\pi - \frac{\pi}{4}, \quad n \in \mathbb{Z} \]