Expression `sin A (1 + tan A) + cos A(1+ cot A)` is equal to:

Options

A. `secA + "cosec"A`
B. `2 "cosec" A (sin A + cos A)`
C. `tan A + cot A`
D. `secA "cosec"A`

Answer

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

### Step 1: Expand the expression
We start by expanding the expression:
\[ \sin A (1 + \tan A) + \cos A (1 + \cot A) = \sin A + \sin A \tan A + \cos A + \cos A \cot A \]

### Step 2: Substitute the definitions of tangent and cotangent
Recall 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:
\[ \sin A + \sin A \left(\frac{\sin A}{\cos A}\right) + \cos A + \cos A \left(\frac{\cos A}{\sin A}\right) \]

### Step 3: Simplify the terms
Now we simplify each term:
\[ \sin A + \frac{\sin^2 A}{\cos A} + \cos A + \frac{\cos^2 A}{\sin A} \]

### Step 4: Combine like terms
We can combine the constant terms:
\[ \sin A + \cos A + \frac{\sin^2 A}{\cos A} + \frac{\cos^2 A}{\sin A} \]

### Step 5: Find a common denominator for the fractions
The common denominator for the fractions is \( \sin A \cos A \):
\[ \sin A + \cos A + \frac{\sin^3 A + \cos^3 A}{\sin A \cos A} \]

### Step 6: Use the identity \( \sin^2 A + \cos^2 A = 1 \)
From the Pythagorean identity, we know that \( \sin^2 A + \cos^2 A = 1 \). Thus:
\[ \sin^3 A + \cos^3 A = (\sin A + \cos A)(\sin^2 A - \sin A \cos A + \cos^2 A) = (\sin A + \cos A)(1 - \sin A \cos A) \]
Substituting this back 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 the expression
This simplifies to:
\[ \sin A + \cos A \left(\frac{\sin A \cos A + 1 - \sin A \cos A}{\sin A \cos A}\right) = \sin A + \cos A \left(\frac{1}{\sin A \cos A}\right) \]

### Final Step: Write the final expression
Thus, the final expression simplifies to:
\[ \sin A + \cos A \left(\frac{1}{\sin A \cos A}\right) = \sin A + \frac{1}{\sin A} + \frac{1}{\cos A} \]

### Conclusion
The expression \( \sin A (1 + \tan A) + \cos A (1 + \cot A) \) simplifies to \( \sec A + \csc A \).