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The problem in your image involves verifying trigonometric identities by utilizing the periodic properties of trigonometric functions.

### Solution:
1. **Identify the Periodicity:**
   - The problem states that \( \sin(t + 2\pi) = \sin(t) \) because the sine function has a period of \( 2\pi \).
   - Similarly, for secant, \( \sec(t + 2\pi) = \sec(t) \).

2. **Rewrite the Expression:**
   - You’re given:
     \[
     \sec(t + 2\pi) \cdot \sin(t + 2\pi) = \sec(t) \cdot \sin(t)
     \]
   - Using the periodic properties, substitute to get:
     \[
     \sec(t) \cdot \sin(t) = \sec(t) \cdot \sin(t)
     \]

3. **Fill in the Blanks:**
   - The blanks should simply be filled with \( \sec(t) \), since:
     \[
     \sec(t + 2\pi) \cdot \sin(t + 2\pi) = \sec(t) \cdot \sin(t)
     \]

The answer for the boxes would likely be:
- First blank: **sec**
- Second blank: **t**

Would you like further explanation on periodic properties, or any steps broken down?

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### Related Questions:
1. What are the periodic properties of other trigonometric functions like cosine or tangent?
2. How would the period change if we modified the argument to something like \( t + \pi \)?
3. How do we prove that \( \sin(t + 2\pi) = \sin(t) \) formally?
4. What are some common applications of trigonometric identities in solving real-world problems?
5. How does secant's periodic property compare with sine’s?

### Tip:
Always remember that adding \(2\pi\) (a full circle) to the angle of sine, cosine, secant, or cosecant functions results in the same value due to their periodicity.

Verify trigonometric identities using the periodic properties of trigonometric functions in the expression sec(t + 2π) ⋅ sin(t + 2π) = sec(t) ⋅ sin(t).

Explore how to verify the trigonometric identity sec(t + 2π) ⋅ sin(t + 2π) = sec(t) ⋅ sin(t) by using periodic properties of trigonometric functions. This example illustrates the periodicity of sine and secant functions and is ideal for students in Grades 10-12.

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