The value of 5 `cot^(2)A-"5 cosec"^(2)A` is :

Options

A. `-1`
B. 5
C. 5
D. 0

Answer

To solve the problem \( 5 \cot^2 A - 5 \csc^2 A \), we can follow these steps:

### Step 1: Factor out the common term
We can factor out the common term \( 5 \):
\[ 5 (\cot^2 A - \csc^2 A) \]

### Step 2: Use the Pythagorean identity
Recall the identity:
\[ \csc^2 A = 1 + \cot^2 A \]
We can substitute this identity into our expression:
\[ \cot^2 A - \csc^2 A = \cot^2 A - (1 + \cot^2 A) \]

### Step 3: Simplify the expression
Now, simplify the expression:
\[ \cot^2 A - \csc^2 A = \cot^2 A - 1 - \cot^2 A = -1 \]

### Step 4: Multiply by 5
Now, substitute back into our factored expression:
\[ 5 (\cot^2 A - \csc^2 A) = 5 \times (-1) = -5 \]

### Final Answer
Thus, the value of \( 5 \cot^2 A - 5 \csc^2 A \) is:
\[ \boxed{-5} \]

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