What is the simplified value of `("cosec"^4 A - cot^2 A )-(cot^4 A+ "cosec"^2A)` ?

Options

A. 0
B. 5
C. 6
D. 9

Answer

To simplify the expression \( \left( \csc^4 A - \cot^2 A \right) - \left( \cot^4 A + \csc^2 A \right) \), we will follow these steps:

### Step 1: Rewrite the expression
Start by rewriting the expression for clarity:
\[ \csc^4 A - \cot^2 A - \cot^4 A - \csc^2 A \]

### Step 2: Group similar terms
Group the terms involving \(\csc\) and \(\cot\):
\[ (\csc^4 A - \csc^2 A) - (\cot^4 A + \cot^2 A) \]

### Step 3: Factor out common terms
Factor out \(\csc^2 A\) from the first group and \(\cot^2 A\) from the second group:
\[ \csc^2 A (\csc^2 A - 1) - \cot^2 A (\cot^2 A + 1) \]

### Step 4: Use Pythagorean identities
Recall the Pythagorean identities:
\[ \csc^2 A = 1 + \cot^2 A \]
Thus, we can substitute \(\csc^2 A - 1\) with \(\cot^2 A\):
\[ \csc^2 A \cdot \cot^2 A - \cot^2 A (\cot^2 A + 1) \]

### Step 5: Simplify the expression
Now we can simplify the expression:
\[ \cot^2 A \cdot \csc^2 A - \cot^4 A - \cot^2 A \]
This can be rewritten as:
\[ \cot^2 A (\csc^2 A - \cot^2 A - 1) \]

### Step 6: Substitute \(\csc^2 A\)
Substituting \(\csc^2 A = 1 + \cot^2 A\):
\[ \cot^2 A \left((1 + \cot^2 A) - \cot^2 A - 1\right) \]
This simplifies to:
\[ \cot^2 A (0) = 0 \]

### Final Answer
Thus, the simplified value of the expression is:
\[ \boxed{0} \]