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

Options

A. 5
B. 2
C. 3
D. 4

Answer

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

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

Thus, we can rewrite \( 1 + \cot^2 \theta \) as:
\[ 1 + \cot^2 \theta = 1 + \frac{\cos^2 \theta}{\sin^2 \theta} = \frac{\sin^2 \theta + \cos^2 \theta}{\sin^2 \theta} = \frac{1}{\sin^2 \theta} \]

Now substituting these into our expression:
\[ \frac{5}{\sec^2 \theta} = 5 \cos^2 \theta \]
\[ \frac{2}{1 + \cot^2 \theta} = 2 \sin^2 \theta \]

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

### Step 4: Use the Pythagorean identity
Using the Pythagorean identity \( \cos^2 \theta + \sin^2 \theta = 1 \):
\[ 5 \cdot 1 = 5 \]

### Final Answer
The numerical value of the expression is \( 5 \).

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