If `tan2theta =1`, then the general value of `theta ` is

Options

A. `(4n + 1) pi/8`
B. `(n+ 1/2 )pi`
C. `npi + 1/2`
D. none of these

Answer

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

### Step 1: Identify the angle whose tangent is 1
The tangent function equals 1 at specific angles. We know that:
\[ \tan \frac{\pi}{4} = 1 \]
Thus, we can write:
\[ 2\theta = \frac{\pi}{4} + n\pi \]
where \( n \) is any integer (since the tangent function has a period of \( \pi \)).

### Step 2: Solve for \( \theta \)
Now, we can solve for \( \theta \) by dividing both sides of the equation by 2:
\[ \theta = \frac{\frac{\pi}{4} + n\pi}{2} \]
This simplifies to:
\[ \theta = \frac{\pi}{8} + \frac{n\pi}{2} \]

### Step 3: Write the general solution
The general solution for \( \theta \) can be expressed as:
\[ \theta = \frac{\pi}{8} + \frac{n\pi}{2}, \quad n \in \mathbb{Z} \]

### Final Answer
Thus, the general value of \( \theta \) is:
\[ \theta = \frac{\pi}{8} + \frac{n\pi}{2}, \quad n \in \mathbb{Z} \]

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