Find the value of `9cosec^2A-9cot^2A`.

Options

A. 1
B. 0
C. `-1`
D. 9

Answer

To find the value of \( 9 \csc^2 A - 9 \cot^2 A \), we can follow these steps:

### Step 1: Rewrite the trigonometric functions
We know that:
\[ \csc^2 A = \frac{1}{\sin^2 A} \quad \text{and} \quad \cot^2 A = \frac{\cos^2 A}{\sin^2 A} \]
Substituting these into the expression gives:
\[ 9 \csc^2 A - 9 \cot^2 A = 9 \left( \frac{1}{\sin^2 A} \right) - 9 \left( \frac{\cos^2 A}{\sin^2 A} \right) \]

### Step 2: Combine the terms
Factoring out \( 9 \) and combining the fractions:
\[ = 9 \left( \frac{1 - \cos^2 A}{\sin^2 A} \right) \]

### Step 3: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ \sin^2 A + \cos^2 A = 1 \implies 1 - \cos^2 A = \sin^2 A \]
Substituting this into the expression gives:
\[ = 9 \left( \frac{\sin^2 A}{\sin^2 A} \right) \]

### Step 4: Simplify the expression
Since \( \frac{\sin^2 A}{\sin^2 A} = 1 \):
\[ = 9 \cdot 1 = 9 \]

### Final Answer
Thus, the value of \( 9 \csc^2 A - 9 \cot^2 A \) is:
\[ \boxed{9} \]