The numerical value of `(1)/(1+cot^(2)theta)+(3)/(1+tan^(2)theta)+2 sin^(2)theta` will be

Options

A. 2
B. 5
C. 6
D. 3

Answer

To solve the expression \( \frac{1}{1 + \cot^2 \theta} + \frac{3}{1 + \tan^2 \theta} + 2 \sin^2 \theta \), we will use trigonometric identities to simplify it step by step.

### Step 1: Use Trigonometric Identities
Recall the following identities:
- \( 1 + \cot^2 \theta = \csc^2 \theta \)
- \( 1 + \tan^2 \theta = \sec^2 \theta \)

Using these identities, we can rewrite the expression:
\[ \frac{1}{1 + \cot^2 \theta} = \frac{1}{\csc^2 \theta} = \sin^2 \theta \]
\[ \frac{3}{1 + \tan^2 \theta} = \frac{3}{\sec^2 \theta} = 3 \cos^2 \theta \]

### Step 2: Substitute Back into the Expression
Now, substituting these identities back into the original expression:
\[ \sin^2 \theta + 3 \cos^2 \theta + 2 \sin^2 \theta \]

### Step 3: Combine Like Terms
Combine the terms involving \( \sin^2 \theta \):
\[ (1 + 2) \sin^2 \theta + 3 \cos^2 \theta = 3 \sin^2 \theta + 3 \cos^2 \theta \]

### Step 4: Factor Out Common Terms
We can factor out the common factor of 3:
\[ 3 (\sin^2 \theta + \cos^2 \theta) \]

### Step 5: Apply the Pythagorean Identity
Using the identity \( \sin^2 \theta + \cos^2 \theta = 1 \):
\[ 3 \cdot 1 = 3 \]

### Conclusion
Thus, the numerical value of the given expression is:
\[ \boxed{3} \]