`4tan^(2)A - 4sec^(2)A` is equal to:

Options

A. 2
B. 3
C. 4
D. `-4`

Answer

To solve the expression \(4\tan^2 A - 4\sec^2 A\), we can follow these steps:

### Step 1: Factor out the common term
We start by factoring out the common factor of 4 from the expression.

\[ 4\tan^2 A - 4\sec^2 A = 4(\tan^2 A - \sec^2 A) \]

### Step 2: Use the trigonometric identity
We know from trigonometric identities that:

\[ \sec^2 A - \tan^2 A = 1 \]

This can be rearranged to:

\[ \tan^2 A - \sec^2 A = -1 \]

### Step 3: Substitute the identity into the expression
Now we can substitute \(-1\) for \(\tan^2 A - \sec^2 A\) in our factored expression:

\[ 4(\tan^2 A - \sec^2 A) = 4(-1) \]

### Step 4: Simplify the expression
Now we simplify the expression:

\[ 4(-1) = -4 \]

### Final Answer
Thus, the value of \(4\tan^2 A - 4\sec^2 A\) is:

\[ \boxed{-4} \]

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