If `costheta=(4)/(5),sectheta+tantheta=?`

Options

A. 2
B. 1
C. 3
D. 4

Answer

To solve the problem, we need to find the value of \( \sec \theta + \tan \theta \) given that \( \cos \theta = \frac{4}{5} \).

### Step-by-Step Solution:

1. **Understanding the relationship**: We know that:
\[ \sec \theta = \frac{1}{\cos \theta} \]
and
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \]

2. **Calculate \( \sec \theta \)**:
Since \( \cos \theta = \frac{4}{5} \), we can find \( \sec \theta \):
\[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{4}{5}} = \frac{5}{4} \]

3. **Finding \( \sin \theta \)**:
To find \( \tan \theta \), we first need \( \sin \theta \). We can use the Pythagorean theorem:
\[ \sin^2 \theta + \cos^2 \theta = 1 \]
Plugging in \( \cos \theta \):
\[ \sin^2 \theta + \left(\frac{4}{5}\right)^2 = 1 \]
\[ \sin^2 \theta + \frac{16}{25} = 1 \]
\[ \sin^2 \theta = 1 - \frac{16}{25} = \frac{25}{25} - \frac{16}{25} = \frac{9}{25} \]
Taking the square root:
\[ \sin \theta = \sqrt{\frac{9}{25}} = \frac{3}{5} \]

4. **Calculate \( \tan \theta \)**:
Now we can find \( \tan \theta \):
\[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{\frac{3}{5}}{\frac{4}{5}} = \frac{3}{4} \]

5. **Combine \( \sec \theta \) and \( \tan \theta \)**:
Now we can find \( \sec \theta + \tan \theta \):
\[ \sec \theta + \tan \theta = \frac{5}{4} + \frac{3}{4} = \frac{5 + 3}{4} = \frac{8}{4} = 2 \]

### Final Answer:
Thus, \( \sec \theta + \tan \theta = 2 \).