Solve `2 tan^(2)x + sec^(2) x= 2`

Answer

To solve the equation \( 2 \tan^2 x + \sec^2 x = 2 \), we will follow these steps:

### Step 1: Rewrite the equation using the identity for secant
We know that:
\[ \sec^2 x = 1 + \tan^2 x \]
Substituting this into the equation gives:
\[ 2 \tan^2 x + (1 + \tan^2 x) = 2 \]

### Step 2: Simplify the equation
Now, simplify the equation:
\[ 2 \tan^2 x + 1 + \tan^2 x = 2 \]
Combine like terms:
\[ 3 \tan^2 x + 1 = 2 \]

### Step 3: Isolate the term with \(\tan^2 x\)
Subtract 1 from both sides:
\[ 3 \tan^2 x = 1 \]

### Step 4: Solve for \(\tan^2 x\)
Divide both sides by 3:
\[ \tan^2 x = \frac{1}{3} \]

### Step 5: Take the square root
Taking the square root of both sides gives:
\[ \tan x = \pm \frac{1}{\sqrt{3}} \]

### Step 6: Find the general solution for \(x\)
The angles for which \(\tan x = \frac{1}{\sqrt{3}}\) are:
\[ x = \frac{\pi}{6} + n\pi \quad \text{and} \quad x = -\frac{\pi}{6} + n\pi \]
where \(n\) is any integer.

### Final Answer
Thus, the general solution for \(x\) is:
\[ x = n\pi \pm \frac{\pi}{6}, \quad n \in \mathbb{Z} \]

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