Find the value of `(1-cosA)/(1+cosA)`.

Options

A. `((1-cosA)/(sinA))^(2)`
B. `(1-cosA)/(2sinA)`
C. `((sinA)/(1-cosA))^(2)`
D. `(sinA)/(1+cosA)`

Answer

To find the value of \(\frac{1 - \cos A}{1 + \cos A}\), we can use trigonometric identities. Let's go through the steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ \frac{1 - \cos A}{1 + \cos A} \]

### Step 2: Multiply numerator and denominator by \(1 + \cos A\)
To simplify, we can multiply both the numerator and the denominator by \(1 + \cos A\):
\[ \frac{(1 - \cos A)(1 + \cos A)}{(1 + \cos A)(1 + \cos A)} = \frac{1 - \cos^2 A}{(1 + \cos A)^2} \]

### Step 3: Use the Pythagorean identity
We know from the Pythagorean identity that:
\[ 1 - \cos^2 A = \sin^2 A \]
Substituting this into our expression gives:
\[ \frac{\sin^2 A}{(1 + \cos A)^2} \]

### Step 4: Final expression
Thus, the value of \(\frac{1 - \cos A}{1 + \cos A}\) simplifies to:
\[ \frac{\sin^2 A}{(1 + \cos A)^2} \]

### Summary
The final result is:
\[ \frac{1 - \cos A}{1 + \cos A} = \frac{\sin^2 A}{(1 + \cos A)^2} \]

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