The numerical value of `(5)/(sec^(2)theta)+(2)/(1+cot^(2)theta)+3sin^(2)theta` is :

Options

A. a)5
B. b)2
C. c)3
D. d)4

Answer

To solve the expression \( \frac{5}{\sec^2 \theta} + \frac{2}{1 + \cot^2 \theta} + 3 \sin^2 \theta \), we will follow these steps:

### Step 1: Rewrite the terms using trigonometric identities
We know that:
- \( \sec^2 \theta = \frac{1}{\cos^2 \theta} \)
- \( 1 + \cot^2 \theta = \csc^2 \theta = \frac{1}{\sin^2 \theta} \)

Using these identities, we can rewrite the expression:

\[ \frac{5}{\sec^2 \theta} = 5 \cos^2 \theta \]
\[ \frac{2}{1 + \cot^2 \theta} = 2 \sin^2 \theta \]

So, the expression becomes:

\[ 5 \cos^2 \theta + 2 \sin^2 \theta + 3 \sin^2 \theta \]

### Step 2: Combine like terms
Now, combine the \( \sin^2 \theta \) terms:

\[ 5 \cos^2 \theta + (2 \sin^2 \theta + 3 \sin^2 \theta) = 5 \cos^2 \theta + 5 \sin^2 \theta \]

### Step 3: Factor out the common term
We can factor out the common factor of 5:

\[ 5 (\cos^2 \theta + \sin^2 \theta) \]

### Step 4: Use the Pythagorean identity
Using the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \):

\[ 5 (\cos^2 \theta + \sin^2 \theta) = 5 \cdot 1 = 5 \]

### Final Answer
Thus, the numerical value of the expression is:

\[ \boxed{5} \]