If `cos theta=(5)/(13)`. Then the value of `tan^2theta+sec^2 theta` is equal to:

Options

A. `(313)/(25)`
B. `(323)/(25)`
C. `(233)/(25)`
D. `(303)/(25)`

Answer

To solve the problem where \( \cos \theta = \frac{5}{13} \) and we need to find the value of \( \tan^2 \theta + \sec^2 \theta \), we can follow these steps:

### Step 1: Identify the values of the triangle
Given \( \cos \theta = \frac{5}{13} \), we can interpret this in the context of a right triangle. Here, the adjacent side (base) is 5 and the hypotenuse is 13.

### Step 2: Use Pythagorean theorem to find the opposite side
We can find the length of the opposite side (perpendicular) using the Pythagorean theorem:

\[ \text{hypotenuse}^2 = \text{base}^2 + \text{perpendicular}^2 \]

Substituting the known values:

\[ 13^2 = 5^2 + \text{perpendicular}^2 \]

Calculating the squares:

\[ 169 = 25 + \text{perpendicular}^2 \]

Now, isolate the perpendicular:

\[ \text{perpendicular}^2 = 169 - 25 = 144 \]

Taking the square root:

\[ \text{perpendicular} = \sqrt{144} = 12 \]

### Step 3: Calculate \( \tan \theta \)
Now that we have the lengths of all sides, we can calculate \( \tan \theta \):

\[ \tan \theta = \frac{\text{perpendicular}}{\text{base}} = \frac{12}{5} \]

### Step 4: Calculate \( \sec \theta \)
Next, we calculate \( \sec \theta \):

\[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{5}{13}} = \frac{13}{5} \]

### Step 5: Calculate \( \tan^2 \theta \) and \( \sec^2 \theta \)
Now we can find \( \tan^2 \theta \) and \( \sec^2 \theta \):

\[ \tan^2 \theta = \left(\frac{12}{5}\right)^2 = \frac{144}{25} \]

\[ \sec^2 \theta = \left(\frac{13}{5}\right)^2 = \frac{169}{25} \]

### Step 6: Add \( \tan^2 \theta \) and \( \sec^2 \theta \)
Now we add these two results:

\[ \tan^2 \theta + \sec^2 \theta = \frac{144}{25} + \frac{169}{25} = \frac{144 + 169}{25} = \frac{313}{25} \]

### Final Answer
Thus, the value of \( \tan^2 \theta + \sec^2 \theta \) is:

\[ \frac{313}{25} \]