What is the simplified value of [(1+sec`2theta)tan^(2)theta]+1` ?

Options

A. `cos2theta`
B. 1
C. `sec2theta`
D. `cosectheta`

Answer

To simplify the expression \([(1 + \sec^2 \theta) \tan^2 \theta] + 1\), we can follow these steps:

### Step 1: Recall the Pythagorean Identity
We know from trigonometric identities that:
\[ \sec^2 \theta = 1 + \tan^2 \theta \]
This identity will help us simplify the expression.

### Step 2: Substitute \(\sec^2 \theta\)
Now, we can substitute \(\sec^2 \theta\) in the expression:
\[ 1 + \sec^2 \theta = 1 + (1 + \tan^2 \theta) = 2 + \tan^2 \theta \]

### Step 3: Rewrite the Expression
Now, we can rewrite the original expression:
\[ (1 + \sec^2 \theta) \tan^2 \theta = (2 + \tan^2 \theta) \tan^2 \theta \]

### Step 4: Expand the Expression
Now, expand the expression:
\[ (2 + \tan^2 \theta) \tan^2 \theta = 2 \tan^2 \theta + \tan^4 \theta \]

### Step 5: Add 1 to the Expression
Now, add 1 to the expanded expression:
\[ 2 \tan^2 \theta + \tan^4 \theta + 1 \]

### Step 6: Rearranging
Rearranging the terms gives us:
\[ \tan^4 \theta + 2 \tan^2 \theta + 1 \]

### Step 7: Factor the Expression
Notice that the expression can be factored:
\[ (\tan^2 \theta + 1)^2 \]

### Final Result
Thus, the simplified value of the original expression is:
\[ (\tan^2 \theta + 1)^2 \]