Find the value of `(1+tan^(2)theta)/(1+cot^(2)theta)`

Answer

To find the value of \(\frac{1 + \tan^2 \theta}{1 + \cot^2 \theta}\), we can follow these steps:

### Step 1: Rewrite cotangent in terms of tangent
We know that \(\cot \theta = \frac{1}{\tan \theta}\). Therefore, we can express \(\cot^2 \theta\) as:
\[ \cot^2 \theta = \frac{1}{\tan^2 \theta} \]

### Step 2: Substitute cotangent in the expression
Now, we can substitute \(\cot^2 \theta\) in the original expression:
\[ \frac{1 + \tan^2 \theta}{1 + \cot^2 \theta} = \frac{1 + \tan^2 \theta}{1 + \frac{1}{\tan^2 \theta}} \]

### Step 3: Simplify the denominator
To simplify the denominator, we need to find a common denominator:
\[ 1 + \frac{1}{\tan^2 \theta} = \frac{\tan^2 \theta + 1}{\tan^2 \theta} \]

### Step 4: Rewrite the entire expression
Now we can rewrite the entire expression:
\[ \frac{1 + \tan^2 \theta}{1 + \cot^2 \theta} = \frac{1 + \tan^2 \theta}{\frac{\tan^2 \theta + 1}{\tan^2 \theta}} = (1 + \tan^2 \theta) \cdot \frac{\tan^2 \theta}{\tan^2 \theta + 1} \]

### Step 5: Cancel out the common terms
Notice that \(1 + \tan^2 \theta\) in the numerator and \(\tan^2 \theta + 1\) in the denominator are the same. Thus, they cancel each other out:
\[ = \tan^2 \theta \]

### Final Answer
The value of \(\frac{1 + \tan^2 \theta}{1 + \cot^2 \theta}\) is:
\[ \tan^2 \theta \]