What is the simplified value of `cosec^(6)A-cot^(6)A-3cosec^(2)Acot^(2)A`?

Options

A. `-2`
B. `-1`
C. 0
D. 1

Answer

To simplify the expression \( \csc^6 A - \cot^6 A - 3 \csc^2 A \cot^2 A \), we can follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \csc^6 A - \cot^6 A - 3 \csc^2 A \cot^2 A \]

### Step 2: Recognize the identity
We know that:
\[ \csc^2 A - \cot^2 A = 1 \]
This identity will be useful in our simplification.

### Step 3: Factor the expression
We can rewrite \( \csc^6 A - \cot^6 A \) using the difference of cubes:
\[ \csc^6 A - \cot^6 A = (\csc^2 A)^3 - (\cot^2 A)^3 \]
Using the formula for the difference of cubes \( a^3 - b^3 = (a - b)(a^2 + ab + b^2) \), we have:
\[ = (\csc^2 A - \cot^2 A)(\csc^4 A + \csc^2 A \cot^2 A + \cot^4 A) \]

### Step 4: Substitute the identity
Substituting the identity \( \csc^2 A - \cot^2 A = 1 \) into the expression gives us:
\[ = 1 \cdot (\csc^4 A + \csc^2 A \cot^2 A + \cot^4 A) \]
Thus, we can simplify our expression to:
\[ \csc^4 A + \csc^2 A \cot^2 A + \cot^4 A - 3 \csc^2 A \cot^2 A \]

### Step 5: Combine like terms
Now, we combine the terms:
\[ \csc^4 A + (\csc^2 A \cot^2 A - 3 \csc^2 A \cot^2 A) + \cot^4 A \]
This simplifies to:
\[ \csc^4 A - 2 \csc^2 A \cot^2 A + \cot^4 A \]

### Step 6: Recognize the square
Notice that:
\[ \csc^4 A - 2 \csc^2 A \cot^2 A + \cot^4 A = (\csc^2 A - \cot^2 A)^2 \]
Using the identity again:
\[ = 1^2 = 1 \]

### Final Answer
Thus, the simplified value of \( \csc^6 A - \cot^6 A - 3 \csc^2 A \cot^2 A \) is:
\[ \boxed{1} \]