The value of `(1/((1 + tan^2 theta)) + 1/((1 + cot^2 theta)))` is

Options

A. 1
B. 2
C. `1/2`
D. `1/4`

Answer

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

### Step 1: Use the Pythagorean Identity
Recall the Pythagorean identity:
\[ 1 + \tan^2 \theta = \sec^2 \theta \]
and
\[ 1 + \cot^2 \theta = \csc^2 \theta \]

### Step 2: Rewrite the Expression
Using the identities from Step 1, we can rewrite the expression:
\[ \frac{1}{1 + \tan^2 \theta} = \frac{1}{\sec^2 \theta} = \cos^2 \theta \]
\[ \frac{1}{1 + \cot^2 \theta} = \frac{1}{\csc^2 \theta} = \sin^2 \theta \]
Thus, the expression becomes:
\[ \cos^2 \theta + \sin^2 \theta \]

### Step 3: Apply the Fundamental Identity
Using the fundamental trigonometric identity:
\[ \cos^2 \theta + \sin^2 \theta = 1 \]

### Conclusion
Therefore, the value of the expression \( \frac{1}{1 + \tan^2 \theta} + \frac{1}{1 + \cot^2 \theta} \) is:
\[ \boxed{1} \]