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Let's analyze the trigonometric problems provided in the image:

### 1. Problem Analysis

The task is to find the exact values of the trigonometric expressions using their periodic properties.

#### a) \( \sec 420^\circ \)

The secant function, \( \sec(x) \), is periodic with a period of \( 360^\circ \). To find \( \sec(420^\circ) \), we can first reduce it within the range of \( 0^\circ \) to \( 360^\circ \):

\[
420^\circ - 360^\circ = 60^\circ
\]

Thus, \( \sec(420^\circ) = \sec(60^\circ) \).

Now, we know that:

\[
\sec(60^\circ) = \frac{1}{\cos(60^\circ)}
\]

And:

\[
\cos(60^\circ) = \frac{1}{2}
\]

So:

\[
\sec(60^\circ) = \frac{1}{\frac{1}{2}} = 2
\]

Therefore:

\[
\sec(420^\circ) = 2
\]

#### b) \( \cot \left( \frac{17\pi}{4} \right) \)

The cotangent function, \( \cot(x) \), is periodic with a period of \( \pi \). To find \( \cot \left( \frac{17\pi}{4} \right) \), we first reduce it within the range of \( 0 \) to \( \pi \):

\[
\frac{17\pi}{4} - 4\pi = \frac{17\pi - 16\pi}{4} = \frac{\pi}{4}
\]

Thus:

\[
\cot \left( \frac{17\pi}{4} \right) = \cot \left( \frac{\pi}{4} \right)
\]

Now, we know:

\[
\cot \left( \frac{\pi}{4} \right) = 1
\]

Therefore:

\[
\cot \left( \frac{17\pi}{4} \right) = 1
\]

### Summary of Answers:

a) \( \sec 420^\circ = 2 \)

b) \( \cot \left( \frac{17\pi}{4} \right) = 1 \)

Would you like to go into more details about how these solutions were derived, or do you have any other questions?

### Related Questions:

1. What is the period of the tangent and cotangent functions?
2. How do you find the exact values of trigonometric functions at specific angles?
3. What are the co-function identities for sine and cosine?
4. How do you determine the signs of trigonometric functions in different quadrants?
5. How are trigonometric functions related to the unit circle?

#### Tip:

When working with trigonometric functions, always remember to reduce angles using their periods to find equivalent values within a standard range (e.g., \(0^\circ\) to \(360^\circ\) or \(0\) to \(2\pi\)).

Use the fact that the trigonometric functions are periodic to find the exact value of each expression: a) sec(420°), b) cot(17π/4).

Find the exact values of sec(420°) and cot(17π/4) by using the periodic properties of trigonometric functions. This problem demonstrates how to reduce angles within standard ranges and apply key trigonometric formulas. Suitable for high school students in Grades 10-12.

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