`sin^(2)Acot^(2)A+cos^(2)Atan^(2)A=?`

Options

A. `-1`
B. 0
C. 3
D. 1

Answer

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

### Step 1: Rewrite cotangent and tangent in terms of sine and cosine
We know that:
\[ \cot A = \frac{\cos A}{\sin A} \quad \text{and} \quad \tan A = \frac{\sin A}{\cos A} \]
Thus, we can express \( \cot^2 A \) and \( \tan^2 A \) as:
\[ \cot^2 A = \left(\frac{\cos A}{\sin A}\right)^2 = \frac{\cos^2 A}{\sin^2 A} \]
\[ \tan^2 A = \left(\frac{\sin A}{\cos A}\right)^2 = \frac{\sin^2 A}{\cos^2 A} \]

### Step 2: Substitute these expressions into the original equation
Now substitute \( \cot^2 A \) and \( \tan^2 A \) into the expression:
\[ \sin^2 A \cot^2 A + \cos^2 A \tan^2 A = \sin^2 A \left(\frac{\cos^2 A}{\sin^2 A}\right) + \cos^2 A \left(\frac{\sin^2 A}{\cos^2 A}\right) \]

### Step 3: Simplify the expression
This simplifies to:
\[ \sin^2 A \cdot \frac{\cos^2 A}{\sin^2 A} + \cos^2 A \cdot \frac{\sin^2 A}{\cos^2 A} = \cos^2 A + \sin^2 A \]

### Step 4: Use the Pythagorean identity
From the Pythagorean identity, we know that:
\[ \sin^2 A + \cos^2 A = 1 \]

### Final Result
Thus, we conclude:
\[ \sin^2 A \cot^2 A + \cos^2 A \tan^2 A = 1 \]