If `sectheta+tantheta=k` find the value of `costheta`

Answer

To solve the problem where \( \sec \theta + \tan \theta = k \) and we need to find the value of \( \cos \theta \), we can follow these steps:

### Step-by-Step Solution:

1. **Start with the given equation:**
\[ \sec \theta + \tan \theta = k \]

2. **Use the identity:**
We know that:
\[ \sec^2 \theta - \tan^2 \theta = 1 \]
This can be factored as:
\[ (\sec \theta + \tan \theta)(\sec \theta - \tan \theta) = 1 \]

3. **Substituting the value of \( \sec \theta + \tan \theta \):**
Substitute \( k \) into the identity:
\[ k(\sec \theta - \tan \theta) = 1 \]
From this, we can express \( \sec \theta - \tan \theta \):
\[ \sec \theta - \tan \theta = \frac{1}{k} \]

4. **Now we have two equations:**
\[ \sec \theta + \tan \theta = k \quad (1) \]
\[ \sec \theta - \tan \theta = \frac{1}{k} \quad (2) \]

5. **Add the two equations:**
Adding equations (1) and (2):
\[ (\sec \theta + \tan \theta) + (\sec \theta - \tan \theta) = k + \frac{1}{k} \]
This simplifies to:
\[ 2\sec \theta = k + \frac{1}{k} \]
Therefore:
\[ \sec \theta = \frac{k + \frac{1}{k}}{2} \]

6. **Relate secant to cosine:**
Recall that:
\[ \sec \theta = \frac{1}{\cos \theta} \]
Thus, we can write:
\[ \frac{1}{\cos \theta} = \frac{k + \frac{1}{k}}{2} \]

7. **Solve for \( \cos \theta \):**
Taking the reciprocal gives:
\[ \cos \theta = \frac{2}{k + \frac{1}{k}} \]
To simplify further:
\[ \cos \theta = \frac{2k}{k^2 + 1} \]

### Final Answer:
\[ \cos \theta = \frac{2k}{k^2 + 1} \]