Find the values of the following expression. `(3-4sin^2theta)/cos^2theta+tan^2theta`

Options

A. 1
B. 2
C. 3
D. none of these

Answer

To solve the expression \((3 - 4\sin^2\theta) / \cos^2\theta + \tan^2\theta\), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{3 - 4\sin^2\theta}{\cos^2\theta} + \tan^2\theta \]
We know that \(\tan^2\theta = \frac{\sin^2\theta}{\cos^2\theta}\). Thus, we can rewrite the expression as:
\[ \frac{3 - 4\sin^2\theta}{\cos^2\theta} + \frac{\sin^2\theta}{\cos^2\theta} \]

### Step 2: Combine the fractions
Now, we can combine the two fractions over a common denominator:
\[ \frac{3 - 4\sin^2\theta + \sin^2\theta}{\cos^2\theta} \]
This simplifies to:
\[ \frac{3 - 3\sin^2\theta}{\cos^2\theta} \]

### Step 3: Factor out the common term
We can factor out the common term in the numerator:
\[ \frac{3(1 - \sin^2\theta)}{\cos^2\theta} \]

### Step 4: Use the Pythagorean identity
Using the Pythagorean identity \(1 - \sin^2\theta = \cos^2\theta\), we can substitute:
\[ \frac{3\cos^2\theta}{\cos^2\theta} \]

### Step 5: Simplify the expression
This simplifies to:
\[ 3 \]

### Final Answer
Thus, the value of the expression is:
\[ \boxed{3} \]