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

Options

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

Answer

To solve the expression \( \frac{1}{1 + \tan^2 \theta} + \frac{1}{1 + \cot^2 \theta} \), we can use some trigonometric identities.

### Step-by-step Solution:

1. **Use Trigonometric Identities:**
We know the following identities:
- \( 1 + \tan^2 \theta = \sec^2 \theta \)
- \( 1 + \cot^2 \theta = \csc^2 \theta \)

2. **Rewrite the Expression:**
Using these identities, we can rewrite the expression:
\[ \frac{1}{1 + \tan^2 \theta} = \frac{1}{\sec^2 \theta} \]
and
\[ \frac{1}{1 + \cot^2 \theta} = \frac{1}{\csc^2 \theta} \]

3. **Simplify Using Reciprocal Identities:**
We know that:
\[ \frac{1}{\sec^2 \theta} = \cos^2 \theta \]
and
\[ \frac{1}{\csc^2 \theta} = \sin^2 \theta \]
Thus, we can rewrite the expression as:
\[ \cos^2 \theta + \sin^2 \theta \]

4. **Apply the Pythagorean Identity:**
According to the Pythagorean identity:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]

5. **Final Result:**
Therefore, the value of the original expression is:
\[ \frac{1}{1 + \tan^2 \theta} + \frac{1}{1 + \cot^2 \theta} = 1 \]