`(1+ tan ^(2) A)/( 1 + cot^(2) A)` equals

Options

A. `sec^(2) A `
B. ` - 1 `
C. ` cot^(2) A `
D. ` tan^(2) A `

Answer

To solve the expression \((1 + \tan^2 A)/(1 + \cot^2 A)\), we can follow these steps:

### Step 1: Use Trigonometric Identities
We know from trigonometric identities that:
\[ 1 + \tan^2 A = \sec^2 A \]
and
\[ 1 + \cot^2 A = \csc^2 A \]

### Step 2: Substitute the Identities
Substituting these identities into the expression, we have:
\[ \frac{1 + \tan^2 A}{1 + \cot^2 A} = \frac{\sec^2 A}{\csc^2 A} \]

### Step 3: Rewrite \(\sec^2 A\) and \(\csc^2 A\)
Recall that:
\[ \sec^2 A = \frac{1}{\cos^2 A} \quad \text{and} \quad \csc^2 A = \frac{1}{\sin^2 A} \]
Thus, we can rewrite the expression as:
\[ \frac{\frac{1}{\cos^2 A}}{\frac{1}{\sin^2 A}} = \frac{1}{\cos^2 A} \cdot \frac{\sin^2 A}{1} = \frac{\sin^2 A}{\cos^2 A} \]

### Step 4: Simplify the Expression
The expression \(\frac{\sin^2 A}{\cos^2 A}\) can be simplified to:
\[ \tan^2 A \]

### Final Answer
Thus, we conclude that:
\[ \frac{1 + \tan^2 A}{1 + \cot^2 A} = \tan^2 A \]