Choose one of the following options which is equal to:
`(1+costheta)/(sintheta)`

Options

A. `(sintheta)/(1-costheta)`
B. `(1+sin^(2)theta)/(sintheta costheta)`
C. `(1+2costheta)/(sintheta costheta)`
D. `(1-2sin^(2)theta)/(costheta)`

Answer

To solve the expression \((1 + \cos \theta) / \sin \theta\), we can simplify it step by step.

### Step 1: Multiply by the Conjugate
We start with the expression:
\[ \frac{1 + \cos \theta}{\sin \theta} \]
To simplify this, we can multiply both the numerator and the denominator by the conjugate of the numerator, which is \(1 - \cos \theta\):
\[ \frac{(1 + \cos \theta)(1 - \cos \theta)}{\sin \theta(1 - \cos \theta)} \]

### Step 2: Apply the Difference of Squares
Using the difference of squares formula \(a^2 - b^2\), we simplify the numerator:
\[ (1 + \cos \theta)(1 - \cos \theta) = 1^2 - \cos^2 \theta = 1 - \cos^2 \theta \]
Thus, our expression now looks like:
\[ \frac{1 - \cos^2 \theta}{\sin \theta(1 - \cos \theta)} \]

### Step 3: Use the Pythagorean Identity
According to the Pythagorean identity, we know that:
\[ \sin^2 \theta + \cos^2 \theta = 1 \implies 1 - \cos^2 \theta = \sin^2 \theta \]
Substituting this into our expression gives:
\[ \frac{\sin^2 \theta}{\sin \theta(1 - \cos \theta)} \]

### Step 4: Simplify the Expression
Now, we can simplify the expression:
\[ \frac{\sin^2 \theta}{\sin \theta(1 - \cos \theta)} = \frac{\sin \theta}{1 - \cos \theta} \]

### Final Result
Thus, the simplified form of the given expression \((1 + \cos \theta) / \sin \theta\) is:
\[ \frac{\sin \theta}{1 - \cos \theta} \]