If `sec^(4)theta-sec^(2)theta=3` then the value of `tan^(4)theta+tan^(2)theta` is :

Options

A. 8
B. 4
C. 6
D. 3

Answer

To solve the equation \( \sec^4 \theta - \sec^2 \theta = 3 \) and find the value of \( \tan^4 \theta + \tan^2 \theta \), we can follow these steps:

### Step 1: Rewrite the equation using the identity
We know that:
\[ \sec^2 \theta = 1 + \tan^2 \theta \]
Using this identity, we can express \( \sec^4 \theta \) in terms of \( \tan^2 \theta \):
\[ \sec^4 \theta = (\sec^2 \theta)^2 = (1 + \tan^2 \theta)^2 \]

### Step 2: Substitute into the equation
Now, substituting \( \sec^4 \theta \) into the original equation gives:
\[ (1 + \tan^2 \theta)^2 - (1 + \tan^2 \theta) = 3 \]

### Step 3: Expand the equation
Expanding \( (1 + \tan^2 \theta)^2 \):
\[ 1 + 2\tan^2 \theta + \tan^4 \theta - 1 - \tan^2 \theta = 3 \]
This simplifies to:
\[ \tan^4 \theta + \tan^2 \theta = 3 \]

### Step 4: Final Result
Thus, the value of \( \tan^4 \theta + \tan^2 \theta \) is:
\[ \boxed{3} \]