`tan ""A/2` is equal to

Options

A. `sqrt((1- sin A)/( 1+ sin A))`
B. `sqrt((1+ sin A)/(1- sin A))`
C. `sqrt((1 -cos A)/(1 + cos A))`
D. `sqrt((1 +cos A)/(1- cos A))`

Answer

To find the value of \( \tan \frac{A}{2} \), we can use the half-angle identity for tangent. The half-angle formula for tangent is given by:

\[ \tan \frac{A}{2} = \frac{1 - \cos A}{\sin A} \]

or

\[ \tan \frac{A}{2} = \frac{\sin A}{1 + \cos A} \]

We can derive this step by step:

### Step 1: Use the Half-Angle Identity
We will use the half-angle identity for tangent:
\[ \tan \frac{A}{2} = \frac{\sin A}{1 + \cos A} \]

### Step 2: Substitute Known Values
If we know the values of \( \sin A \) and \( \cos A \), we can substitute them into the formula. For example, if \( \sin A = \frac{3}{5} \) and \( \cos A = \frac{4}{5} \), we substitute these values into the formula.

### Step 3: Calculate
Using the values:
\[ \tan \frac{A}{2} = \frac{\frac{3}{5}}{1 + \frac{4}{5}} = \frac{\frac{3}{5}}{\frac{9}{5}} = \frac{3}{9} = \frac{1}{3} \]

### Final Answer
Thus, \( \tan \frac{A}{2} = \frac{1}{3} \).

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