What the value of sin A cos A tan A + cos A sin A cotA?

Options

A. sin A
B. cos A
C. tan A
D. 1

Answer

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

### Step 1: Write the expression
The given expression is:
\[ \sin A \cos A \tan A + \cos A \sin A \cot A \]

### Step 2: Substitute the definitions of \(\tan A\) and \(\cot A\)
We know that:
\[ \tan A = \frac{\sin A}{\cos A} \quad \text{and} \quad \cot A = \frac{\cos A}{\sin A} \]
Substituting these into the expression gives:
\[ \sin A \cos A \left(\frac{\sin A}{\cos A}\right) + \cos A \sin A \left(\frac{\cos A}{\sin A}\right) \]

### Step 3: Simplify each term
Now, simplify each term:
1. For the first term:
\[ \sin A \cos A \cdot \frac{\sin A}{\cos A} = \sin^2 A \]
2. For the second term:
\[ \cos A \sin A \cdot \frac{\cos A}{\sin A} = \cos^2 A \]

### Step 4: Combine the simplified terms
Now, we can combine the two simplified terms:
\[ \sin^2 A + \cos^2 A \]

### Step 5: Use the Pythagorean identity
According to the Pythagorean identity:
\[ \sin^2 A + \cos^2 A = 1 \]

### Final Answer
Thus, the value of the original expression is:
\[ \boxed{1} \]

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