What is the simplified value of `"cosec"^(6)A-cot^(6)A-3"cosec"^(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: Use Trigonometric Identities
Recall the identities:
- \( \csc^2 A = 1 + \cot^2 A \)

### Step 2: Substitute \( \csc^2 A \)
Let \( x = \cot^2 A \). Then, we can express \( \csc^2 A \) in terms of \( x \):
\[ \csc^2 A = 1 + x \]

### Step 3: Rewrite the Expression
Now, we can rewrite \( \csc^6 A \) and \( \cot^6 A \):
\[ \csc^6 A = (1 + x)^3 \]
\[ \cot^6 A = x^3 \]

### Step 4: Expand \( \csc^6 A \)
Using the binomial theorem:
\[ (1 + x)^3 = 1 + 3x + 3x^2 + x^3 \]

### Step 5: Substitute Back into the Expression
Now substitute back into the original expression:
\[ \csc^6 A - \cot^6 A - 3 \csc^2 A \cot^2 A = (1 + 3x + 3x^2 + x^3) - x^3 - 3(1 + x)x \]

### Step 6: Simplify the Expression
Now simplify:
\[ = 1 + 3x + 3x^2 + x^3 - x^3 - 3(1 + x)x \]
\[ = 1 + 3x + 3x^2 - 3x - 3x^2 \]
\[ = 1 + 3x - 3x = 1 \]

### Final Answer
Thus, the simplified value of the expression is:
\[ \boxed{1} \]