Assertion (A) : `((1+cos theta)/(sintheta))^(2)=(1+cos theta)/(1-cos theta)`
Reason (R ) : `sin^(2)theta = cos^(2) theta-1 `

Options

A. Both A and R are true and R is the correct explanation of A.
B. Both A and R are true and R is NOT the correct explanation of A.
C. A is true but R is false
D. A is false but R is true

Answer

To solve the assertion and reason problem, we need to analyze the given statements step by step.

### Step 1: Analyze the Assertion
The assertion states:
\[ \left(\frac{1 + \cos \theta}{\sin \theta}\right)^2 = \frac{1 + \cos \theta}{1 - \cos \theta} \]

**Left Hand Side (LHS)**:
\[ \left(\frac{1 + \cos \theta}{\sin \theta}\right)^2 = \frac{(1 + \cos \theta)^2}{\sin^2 \theta} \]

**Right Hand Side (RHS)**:
\[ \frac{1 + \cos \theta}{1 - \cos \theta} \]

### Step 2: Simplify the LHS
Using the identity \(\sin^2 \theta + \cos^2 \theta = 1\), we can express \(\sin^2 \theta\) as:
\[ \sin^2 \theta = 1 - \cos^2 \theta \]

Substituting this into the LHS:
\[ \frac{(1 + \cos \theta)^2}{1 - \cos^2 \theta} \]

### Step 3: Factor the Denominator
The denominator \(1 - \cos^2 \theta\) can be factored as:
\[ 1 - \cos^2 \theta = (1 + \cos \theta)(1 - \cos \theta) \]

### Step 4: Substitute Back into LHS
Now substituting this back into the LHS gives:
\[ \frac{(1 + \cos \theta)^2}{(1 + \cos \theta)(1 - \cos \theta)} \]

### Step 5: Cancel Common Terms
Now, we can cancel \(1 + \cos \theta\) from the numerator and denominator (assuming \(1 + \cos \theta \neq 0\)):
\[ \frac{1 + \cos \theta}{1 - \cos \theta} \]

### Conclusion for Assertion
Thus, we have shown that:
\[ \left(\frac{1 + \cos \theta}{\sin \theta}\right)^2 = \frac{1 + \cos \theta}{1 - \cos \theta} \]
This means the assertion (A) is **true**.

### Step 6: Analyze the Reason
The reason states:
\[ \sin^2 \theta = \cos^2 \theta - 1 \]
This is incorrect because the correct identity is:
\[ \sin^2 \theta = 1 - \cos^2 \theta \]
Thus, the reason (R) is **false**.

### Final Answer
- Assertion (A) is **true**.
- Reason (R) is **false**.