Consider the following
(i) `(cos^(2) theta - sin^(2) theta)/(cos^(2) theta + sin^(2) theta) = cos^(2) theta (1 + tan theta)`
(ii) `(1 + sin theta)/(1 - sin theta) = (tan theta + sec theta)^(2)`
Which of the equation given above is/are correct ?

Options

A. Only (i)
B. Only (ii)
C. Both (i) and (ii)
D. Neither (i) nor (ii)

Answer

To determine the correctness of the given statements, we will evaluate each statement step by step.

### Statement (i):
\[ \frac{\cos^2 \theta - \sin^2 \theta}{\cos^2 \theta + \sin^2 \theta} = \cos^2 \theta (1 + \tan \theta) \]

1. **Evaluate the left-hand side (LHS)**:
- We know that \(\cos^2 \theta + \sin^2 \theta = 1\).
- Therefore, the LHS simplifies to:
\[ \frac{\cos^2 \theta - \sin^2 \theta}{1} = \cos^2 \theta - \sin^2 \theta \]

2. **Evaluate the right-hand side (RHS)**:
- We can express \(\tan \theta\) as \(\frac{\sin \theta}{\cos \theta}\).
- Thus, the RHS becomes:
\[ \cos^2 \theta \left(1 + \frac{\sin \theta}{\cos \theta}\right) = \cos^2 \theta + \cos \theta \sin \theta \]

3. **Set LHS equal to RHS**:
- We need to check if:
\[ \cos^2 \theta - \sin^2 \theta = \cos^2 \theta + \cos \theta \sin \theta \]
- Rearranging gives:
\[ -\sin^2 \theta = \cos \theta \sin \theta \]
- This is not generally true, so Statement (i) is **incorrect**.

### Statement (ii):
\[ \frac{1 + \sin \theta}{1 - \sin \theta} = (\tan \theta + \sec \theta)^2 \]

1. **Evaluate the left-hand side (LHS)**:
- The LHS remains as:
\[ \frac{1 + \sin \theta}{1 - \sin \theta} \]

2. **Evaluate the right-hand side (RHS)**:
- We know that \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and \(\sec \theta = \frac{1}{\cos \theta}\).
- Therefore, the RHS becomes:
\[ \left(\frac{\sin \theta}{\cos \theta} + \frac{1}{\cos \theta}\right)^2 = \left(\frac{\sin \theta + 1}{\cos \theta}\right)^2 = \frac{(\sin \theta + 1)^2}{\cos^2 \theta} \]

3. **Set LHS equal to RHS**:
- We need to check if:
\[ \frac{1 + \sin \theta}{1 - \sin \theta} = \frac{(\sin \theta + 1)^2}{\cos^2 \theta} \]
- Cross-multiplying gives:
\[ (1 + \sin \theta) \cos^2 \theta = (1 - \sin \theta)(1 + \sin \theta)^2 \]
- Expanding both sides and simplifying shows that both sides are equal, confirming that Statement (ii) is **correct**.

### Conclusion:
- Statement (i) is incorrect.
- Statement (ii) is correct.

Thus, the correct answer is that only the second statement is correct.