Find the values of the following expression. `sin^2Acot^2A+cos^2Atan^2A`=?

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 will follow these steps:

### Step 1: Rewrite cotangent and tangent in terms of sine and cosine
We know that:
\[ \cot A = \frac{\cos A}{\sin A} \quad \text{and} \quad \tan A = \frac{\sin A}{\cos A} \]
Now, we can substitute these definitions into the expression.

### Step 2: Substitute cotangent and tangent
Substituting the definitions into the expression gives:
\[ \sin^2 A \cot^2 A + \cos^2 A \tan^2 A = \sin^2 A \left(\frac{\cos^2 A}{\sin^2 A}\right) + \cos^2 A \left(\frac{\sin^2 A}{\cos^2 A}\right) \]

### Step 3: Simplify the expression
This simplifies to:
\[ \sin^2 A \cdot \frac{\cos^2 A}{\sin^2 A} + \cos^2 A \cdot \frac{\sin^2 A}{\cos^2 A} \]
The \(\sin^2 A\) in the first term cancels out:
\[ \cos^2 A + \frac{\sin^2 A \cdot \cos^2 A}{\cos^2 A} \]
And the \(\cos^2 A\) in the second term cancels out:
\[ \cos^2 A + \sin^2 A \]

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

### Final Result
Thus, the value of the expression \( \sin^2 A \cot^2 A + \cos^2 A \tan^2 A \) is:
\[ \boxed{1} \]