If `1+ tan^(2) theta = sec^(2) theta` then find `(sec theta + tan theta)`

Answer

To solve the equation \(1 + \tan^2 \theta = \sec^2 \theta\) and find the value of \((\sec \theta + \tan \theta)\), we can follow these steps:

### Step 1: Understand the Given Identity
We start with the identity:
\[ 1 + \tan^2 \theta = \sec^2 \theta \]
This is a well-known trigonometric identity.

### Step 2: Rearranging the Identity
From the identity, we can rearrange it to express \(\sec^2 \theta\) in terms of \(\tan^2 \theta\):
\[ \sec^2 \theta - \tan^2 \theta = 1 \]

### Step 3: Use the Difference of Squares
We can recognize that the left side of the equation can be factored using the difference of squares:
\[ (\sec \theta - \tan \theta)(\sec \theta + \tan \theta) = 1 \]

### Step 4: Solve for \((\sec \theta + \tan \theta)\)
Let \(x = \sec \theta + \tan \theta\) and \(y = \sec \theta - \tan \theta\). From the previous step, we have:
\[ xy = 1 \]
This implies:
\[ x = \frac{1}{y} \]

### Step 5: Find the Value of \((\sec \theta + \tan \theta)\)
To find \(x\), we can express \(y\) in terms of \(x\):
\[ \sec \theta - \tan \theta = \frac{1}{\sec \theta + \tan \theta} \]
Thus, substituting \(x\) into the equation gives us:
\[ y = \frac{1}{x} \]
Now substituting back into the product:
\[ x \cdot \frac{1}{x} = 1 \]
This confirms our identity holds true.

### Conclusion
Thus, we can conclude that:
\[ \sec \theta + \tan \theta = x \]
And since \(xy = 1\), we can say that \((\sec \theta + \tan \theta)\) is the reciprocal of \((\sec \theta - \tan \theta)\).

### Final Result
The value of \((\sec \theta + \tan \theta)\) can be expressed in terms of \(y\) but cannot be simplified further without specific values of \(\theta\).