`4 tan^(2) A - 4 sec^(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: Use the identity for secant
We know from trigonometric identities that:
\[ \sec^2 A = 1 + \tan^2 A \]
We will substitute this identity into our expression.

### Step 2: Substitute the identity into the expression
Substituting \(\sec^2 A\) into the expression, we have:
\[ 4 \tan^2 A - 4 \sec^2 A = 4 \tan^2 A - 4(1 + \tan^2 A) \]

### Step 3: Distribute the -4
Next, we distribute the \(-4\) across the terms in the parentheses:
\[ = 4 \tan^2 A - 4 - 4 \tan^2 A \]

### Step 4: Combine like terms
Now, we can combine the terms:
\[ 4 \tan^2 A - 4 \tan^2 A - 4 = 0 - 4 = -4 \]

### Final Result
Thus, the expression simplifies to:
\[ 4 \tan^2 A - 4 \sec^2 A = -4 \]

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