Answer :

Let's break down the expression step by step:

### Step 1: Understanding the Expression

The expression we need to evaluate is:
[tex]\[ 625 \cdot \sec^2\left(\tan^{-1}\left(\frac{13}{25}\right) \cdot 8\pi\right) \][/tex]

However, it seems like the problem should involve finding the secant squared of a particular angle. Let's focus on evaluating:
[tex]\[ \sec^2\left(\tan^{-1}\left(\frac{13}{25}\right)\right) \][/tex]

### Step 2: Find the Angle

The expression [tex]\(\tan^{-1}\left(\frac{13}{25}\right)\)[/tex] means that we are looking for an angle whose tangent is [tex]\(\frac{13}{25}\)[/tex]. This can be thought of as:

- Opposite side of a right triangle = 13
- Adjacent side of the triangle = 25

### Step 3: Use Trigonometric Identities

Since [tex]\(\sec(\theta) = \frac{1}{\cos(\theta)}\)[/tex], we need [tex]\(\cos(\theta)\)[/tex]:

1. Find the Hypotenuse: Use the Pythagorean theorem.
[tex]\[
\text{Hypotenuse} = \sqrt{13^2 + 25^2} = \sqrt{169 + 625} = \sqrt{794}
\][/tex]

2. Calculate [tex]\(\cos(\theta)\)[/tex]:
[tex]\[
\cos(\theta) = \frac{\text{Adjacent side}}{\text{Hypotenuse}} = \frac{25}{\sqrt{794}}
\][/tex]

3. Calculate [tex]\(\sec(\theta)\)[/tex]:
[tex]\[
\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\sqrt{794}}{25}
\][/tex]

4. Calculate [tex]\(\sec^2(\theta)\)[/tex]:
[tex]\[
\sec^2(\theta) = \left(\frac{\sqrt{794}}{25}\right)^2 = \frac{794}{625}
\][/tex]

### Step 4: Multiply by 625

Finally, we multiply [tex]\(\sec^2(\theta)\)[/tex] by 625:
[tex]\[
625 \cdot \sec^2(\theta) = 625 \cdot \frac{794}{625} = 794
\][/tex]

### Conclusion

The final result of the expression is 794.0.

Thank you for reading the article Evaluate the expression 625 cdot sec 2 left tan 1 left frac 13 25 right right. We hope the information provided is useful and helps you understand this topic better. Feel free to explore more helpful content on our website!

Rewritten by : Jeany

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