Which of the following is /are the value (s) of the the expression ?
sin A (1 + tan A ) + cos A (1 + cot A)
1. sec A + cosec A
2. 2 cosec A (sin a + cos A)
3. tan A + cot A
Select the correct answer using the code given below :

Options

A. 1 only
B. 1 and 2 only
C. 2 only
D. 1 and 3 only

Answer

To solve the expression \( \sin A (1 + \tan A) + \cos A (1 + \cot A) \), we will simplify it step by step.

### Step 1: Rewrite the Tangent and Cotangent
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:
\[ \sin A \left(1 + \frac{\sin A}{\cos A}\right) + \cos A \left(1 + \frac{\cos A}{\sin A}\right) \]

### Step 2: Distribute the Terms
Now, we distribute the terms:
\[ = \sin A + \frac{\sin^2 A}{\cos A} + \cos A + \frac{\cos^2 A}{\sin A} \]

### Step 3: Combine Like Terms
Combine the sine and cosine terms:
\[ = \sin A + \cos A + \frac{\sin^2 A}{\cos A} + \frac{\cos^2 A}{\sin A} \]

### Step 4: Find a Common Denominator
To combine the fractions, we need a common denominator, which is \( \sin A \cos A \):
\[ = \sin A + \cos A + \frac{\sin^3 A + \cos^3 A}{\sin A \cos A} \]

### Step 5: Use the Identity \( \sin^2 A + \cos^2 A = 1 \)
We know that \( \sin^2 A + \cos^2 A = 1 \), so we can rewrite \( \sin^3 A + \cos^3 A \) using the identity:
\[ \sin^3 A + \cos^3 A = (\sin A + \cos A)(\sin^2 A - \sin A \cos A + \cos^2 A) = (\sin A + \cos A)(1 - \sin A \cos A) \]

### Step 6: Substitute Back
Substituting back into the expression gives:
\[ = \sin A + \cos A + \frac{(\sin A + \cos A)(1 - \sin A \cos A)}{\sin A \cos A} \]

### Step 7: Factor Out \( \sin A + \cos A \)
Now, we can factor out \( \sin A + \cos A \):
\[ = (\sin A + \cos A) \left(1 + \frac{1 - \sin A \cos A}{\sin A \cos A}\right) \]

### Step 8: Simplify Further
This simplifies to:
\[ = (\sin A + \cos A) \left(1 + \frac{1}{\sin A \cos A} - 1\right) = (\sin A + \cos A) \left(\frac{1}{\sin A \cos A}\right) \]

### Step 9: Final Expression
Thus, we have:
\[ = \frac{\sin A + \cos A}{\sin A \cos A} \]
This can be rewritten as:
\[ = \csc A + \sec A \]

### Conclusion
So, the expression simplifies to \( \sec A + \csc A \).

### Final Answer:
The correct option is **1. sec A + cosec A**.

---