`sec^(4)alpha(1-sin^(4)alpha)-2tan^(2)alpha=?`

Options

A. a) 1
B. b) 2
C. c) 3
D. d) 0

Answer

To solve the expression \( \sec^4 \alpha (1 - \sin^4 \alpha) - 2 \tan^2 \alpha \), we can break it down step by step.

### Step 1: Rewrite the expression
Start with the original expression:
\[ \sec^4 \alpha (1 - \sin^4 \alpha) - 2 \tan^2 \alpha \]

### Step 2: Factor \( 1 - \sin^4 \alpha \)
Notice that \( 1 - \sin^4 \alpha \) can be factored using the difference of squares:
\[ 1 - \sin^4 \alpha = (1 - \sin^2 \alpha)(1 + \sin^2 \alpha) \]
Since \( 1 - \sin^2 \alpha = \cos^2 \alpha \), we can rewrite it as:
\[ 1 - \sin^4 \alpha = \cos^2 \alpha (1 + \sin^2 \alpha) \]

### Step 3: Substitute back into the expression
Substituting this back into our expression gives:
\[ \sec^4 \alpha (\cos^2 \alpha (1 + \sin^2 \alpha)) - 2 \tan^2 \alpha \]

### Step 4: Simplify \( \sec^4 \alpha \)
Recall that \( \sec \alpha = \frac{1}{\cos \alpha} \), so:
\[ \sec^4 \alpha = \frac{1}{\cos^4 \alpha} \]
Thus, we can rewrite the expression as:
\[ \frac{1}{\cos^4 \alpha} (\cos^2 \alpha (1 + \sin^2 \alpha)) - 2 \tan^2 \alpha \]

### Step 5: Simplify the first term
Now simplify the first term:
\[ \frac{\cos^2 \alpha (1 + \sin^2 \alpha)}{\cos^4 \alpha} = \frac{1 + \sin^2 \alpha}{\cos^2 \alpha} = \sec^2 \alpha (1 + \sin^2 \alpha) \]

### Step 6: Rewrite \( \tan^2 \alpha \)
Recall that \( \tan^2 \alpha = \frac{\sin^2 \alpha}{\cos^2 \alpha} \), so:
\[ 2 \tan^2 \alpha = 2 \frac{\sin^2 \alpha}{\cos^2 \alpha} \]

### Step 7: Combine the terms
Now we have:
\[ \sec^2 \alpha (1 + \sin^2 \alpha) - 2 \frac{\sin^2 \alpha}{\cos^2 \alpha} \]

### Step 8: Use the identity \( \sec^2 \alpha = 1 + \tan^2 \alpha \)
Substituting \( \sec^2 \alpha \) gives:
\[ (1 + \tan^2 \alpha)(1 + \sin^2 \alpha) - 2 \tan^2 \alpha \]

### Step 9: Expand the expression
Expanding this results in:
\[ 1 + \tan^2 \alpha + \sin^2 \alpha + \tan^2 \alpha \sin^2 \alpha - 2 \tan^2 \alpha \]
Combining like terms yields:
\[ 1 + \sin^2 \alpha - \tan^2 \alpha + \tan^2 \alpha \sin^2 \alpha \]

### Step 10: Use the identity \( 1 + \sin^2 \alpha - \tan^2 \alpha = 1 \)
Using the identity \( \sec^2 \alpha - \tan^2 \alpha = 1 \):
\[ 1 + \sin^2 \alpha - \tan^2 \alpha = 1 \]

### Final Answer
Thus, the final result simplifies to:
\[ 1 \]