`8cot ^(2) A-8 cosec ^(2)A ` is equal to

Options

A. 8
B. `1/8`
C. `-8`
D. `-1/8`

Answer

To solve the expression \( 8 \cot^2 A - 8 \csc^2 A \), we can follow these steps:

### Step 1: Rewrite cotangent and cosecant in terms of sine and cosine
We know that:
\[ \cot A = \frac{\cos A}{\sin A} \quad \text{and} \quad \csc A = \frac{1}{\sin A} \]
Thus, we can express \( \cot^2 A \) and \( \csc^2 A \) as:
\[ \cot^2 A = \frac{\cos^2 A}{\sin^2 A} \quad \text{and} \quad \csc^2 A = \frac{1}{\sin^2 A} \]

### Step 2: Substitute these into the expression
Substituting these into the original expression gives:
\[ 8 \cot^2 A - 8 \csc^2 A = 8 \left( \frac{\cos^2 A}{\sin^2 A} \right) - 8 \left( \frac{1}{\sin^2 A} \right) \]

### Step 3: Factor out the common term
We can factor out \( 8 \) and the common denominator \( \sin^2 A \):
\[ = 8 \left( \frac{\cos^2 A - 1}{\sin^2 A} \right) \]

### Step 4: Use the Pythagorean identity
Using the Pythagorean identity \( \sin^2 A + \cos^2 A = 1 \), we can express \( \cos^2 A - 1 \) as:
\[ \cos^2 A - 1 = -\sin^2 A \]
Substituting this back into our expression gives:
\[ = 8 \left( \frac{-\sin^2 A}{\sin^2 A} \right) \]

### Step 5: Simplify the expression
The \( \sin^2 A \) terms cancel out:
\[ = 8 \cdot (-1) = -8 \]

### Final Answer
Thus, the expression \( 8 \cot^2 A - 8 \csc^2 A \) simplifies to:
\[ \boxed{-8} \]