If `(cosA)/(1-sinA)=tan(k+A/2),` then k=

Options

A. `(pi)/2`
B. `-(pi)/4`
C. `(pi)/4`
D. `-(pi)/2`

Answer

To solve the equation \(\frac{\cos A}{1 - \sin A} = \tan\left(k + \frac{A}{2}\right)\), we will follow these steps:

### Step 1: Rewrite the Left-Hand Side (LHS)
We start with the LHS:
\[ \frac{\cos A}{1 - \sin A} \]
Using the identities for cosine and sine in terms of tangent, we can express \(\cos A\) and \(\sin A\) as:
\[ \cos A = \frac{1 - \tan^2\left(\frac{A}{2}\right)}{1 + \tan^2\left(\frac{A}{2}\right)} \]
\[ \sin A = \frac{2\tan\left(\frac{A}{2}\right)}{1 + \tan^2\left(\frac{A}{2}\right)} \]
Substituting these into the LHS gives:
\[ \frac{\frac{1 - \tan^2\left(\frac{A}{2}\right)}{1 + \tan^2\left(\frac{A}{2}\right)}}{1 - \frac{2\tan\left(\frac{A}{2}\right)}{1 + \tan^2\left(\frac{A}{2}\right)}} \]

### Step 2: Simplify the Denominator
The denominator can be simplified:
\[ 1 - \frac{2\tan\left(\frac{A}{2}\right)}{1 + \tan^2\left(\frac{A}{2}\right)} = \frac{(1 + \tan^2\left(\frac{A}{2}\right)) - 2\tan\left(\frac{A}{2}\right)}{1 + \tan^2\left(\frac{A}{2}\right)} = \frac{(1 - \tan\left(\frac{A}{2}\right)^2)}{1 + \tan^2\left(\frac{A}{2}\right)} \]

### Step 3: Combine the Fractions
Now we can combine the fractions:
\[ \frac{1 - \tan^2\left(\frac{A}{2}\right)}{1 + \tan^2\left(\frac{A}{2}\right)} \cdot \frac{1 + \tan^2\left(\frac{A}{2}\right)}{1 - \tan^2\left(\frac{A}{2}\right)} = \frac{(1 - \tan^2\left(\frac{A}{2}\right))^2}{(1 + \tan^2\left(\frac{A}{2}\right))(1 - \tan^2\left(\frac{A}{2}\right))} \]

### Step 4: Use the Tangent Addition Formula
This can be expressed in the form of a tangent function:
\[ \frac{1 + \tan\left(\frac{A}{2}\right)}{1 - \tan\left(\frac{A}{2}\right)} = \tan\left(\frac{\pi}{4} + \frac{A}{2}\right) \]

### Step 5: Set the LHS Equal to the RHS
Now we have:
\[ \tan\left(\frac{\pi}{4} + \frac{A}{2}\right) = \tan\left(k + \frac{A}{2}\right) \]

### Step 6: Compare Angles
Since the tangents are equal, we can set the angles equal to each other:
\[ k + \frac{A}{2} = \frac{\pi}{4} + \frac{A}{2} \]

### Step 7: Solve for \(k\)
Subtract \(\frac{A}{2}\) from both sides:
\[ k = \frac{\pi}{4} \]

### Final Answer
Thus, the value of \(k\) is:
\[ \boxed{\frac{\pi}{4}} \]