What is the value of (tan A + cot A) sin A x cos A ?

Answer

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

### Step 1: Rewrite \(\tan A\) and \(\cot A\)
We know that:
\[ \tan A = \frac{\sin A}{\cos A} \quad \text{and} \quad \cot A = \frac{\cos A}{\sin A} \]
So, we can rewrite the expression as:
\[ (\tan A + \cot A) = \left(\frac{\sin A}{\cos A} + \frac{\cos A}{\sin A}\right) \]

### Step 2: Combine the fractions
To combine the fractions, we need a common denominator, which is \(\sin A \cos A\):
\[ \tan A + \cot A = \frac{\sin^2 A + \cos^2 A}{\sin A \cos A} \]

### Step 3: Substitute back into the expression
Now, we substitute this back into our original expression:
\[ (\tan A + \cot A) \sin A \cos A = \left(\frac{\sin^2 A + \cos^2 A}{\sin A \cos A}\right) \sin A \cos A \]

### Step 4: Simplify the expression
The \(\sin A \cos A\) in the numerator and denominator cancels out:
\[ = \sin^2 A + \cos^2 A \]

### Step 5: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ \sin^2 A + \cos^2 A = 1 \]

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