`cot ^(2) A - cos ^(2) A ` is equal to :

Options

A. `cos ^(2) A - tan ^(2) A`
B. `cot^(2) A`
C. `cos ^(2) A cot ^(2) A`
D. `cos ^(2) A sec ^(2) A`

Answer

To solve the expression \( \cot^2 A - \cos^2 A \), we can follow these steps:

### Step 1: Rewrite \( \cot^2 A \)
We know that:
\[ \cot A = \frac{\cos A}{\sin A} \]
Thus,
\[ \cot^2 A = \frac{\cos^2 A}{\sin^2 A} \]

### Step 2: Substitute \( \cot^2 A \) into the expression
Now we substitute \( \cot^2 A \) into the original expression:
\[ \cot^2 A - \cos^2 A = \frac{\cos^2 A}{\sin^2 A} - \cos^2 A \]

### Step 3: Find a common denominator
To combine the terms, we need a common denominator, which is \( \sin^2 A \):
\[ \frac{\cos^2 A}{\sin^2 A} - \cos^2 A = \frac{\cos^2 A}{\sin^2 A} - \frac{\cos^2 A \sin^2 A}{\sin^2 A} \]
This simplifies to:
\[ \frac{\cos^2 A - \cos^2 A \sin^2 A}{\sin^2 A} \]

### Step 4: Factor out \( \cos^2 A \)
Now we can factor \( \cos^2 A \) from the numerator:
\[ \frac{\cos^2 A (1 - \sin^2 A)}{\sin^2 A} \]

### Step 5: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ 1 - \sin^2 A = \cos^2 A \]
Substituting this into our expression gives:
\[ \frac{\cos^2 A \cdot \cos^2 A}{\sin^2 A} = \frac{\cos^4 A}{\sin^2 A} \]

### Final Result
Thus, the expression \( \cot^2 A - \cos^2 A \) simplifies to:
\[ \frac{\cos^4 A}{\sin^2 A} \]

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