In the image you've provided, we're tasked with several parts of trigonometric problems concerning a right triangle $\triangle XYZ$ where $\angle Z$ is a right angle.

Part 1

Given that $\angle X$ and $\angle Y$ are complementary (since $\angle Z$ is a right angle), $\angle X + \angle Y = 90^\circ$. Thus, the following relationships hold:

• $\sin X = \cos Y$
• $\cos X = \sin Y$

Part 2

Here, we should fill in the trigonometric functions based on the complementarity noted above:

• $\sin X = \cos Y$
• $\cos X = \sin Y$

Part 3

We need to select true statements about the trigonometric functions:

• $\cos X^2 = \cos Y$ is not generally true unless under specific angle conditions.
• $\sin X = \sin Y$ and $\cos X = \sin Y$ might be true depending on the specific angle but not universally so.
• $\cos X = \sin Y$ is correct based on complementary angles.

Part 4

To find $\cos(66^\circ) = \sin(90^\circ - 66^\circ) = \sin(24^\circ)$. Thus, $\cos(66^\circ)$ equals $\sin(24^\circ)$.

If you need further clarification on any of these parts, or if you have more questions on this topic, please let me know! Here are some possible follow-up questions:

1. Can you explain why sine and cosine of complementary angles are equal?
2. What would $\sin X$ and $\cos Y$ be if $X = 30^\circ$?
3. How does the Pythagorean identity apply to this triangle?
4. Could you provide examples of using trigonometric identities in other contexts?
5. How do trigonometric functions apply to non-right triangles?

Tip: Remember, the cosine of any angle in a right triangle is equal to the sine of its complementary angle, which is helpful in simplifying expressions and solving problems.