https://file.aiquickdraw.com/mathbot/file/1730686295209rs9h1pyg.jpg

Let's go through each of these exercises based on the choices given in the image.

The choices available are:
1. \( \frac{\cos^2 x}{\sin^2 x} \)
2. \( \frac{1}{\csc^2 x} \)
3. \( \csc^2 x - \cot^2 x + \sin^2 x \)
4. \( \sin(-x) \)
5. \( \cot x \)

### Problem Analysis and Solutions:

1. **Problem 39: \( - \tan x \cos x = \) ?**
   - Using trigonometric identities, \( \sin(-x) = -\sin x \).
   - Therefore, \( -\tan x \cos x \) should match **\( \sin(-x) \)**.

2. **Problem 40: \( \csc^2 x - 1 = \) ?**
   - We know from the Pythagorean identity \( \csc^2 x - \cot^2 x = 1 \).
   - Rearranging gives \( \csc^2 x - 1 = \cot^2 x \).
   - The answer is **\( \frac{\cos^2 x}{\sin^2 x} \)**.

3. **Problem 41: \( \frac{\csc x}{\sec x} = \) ?**
   - Simplifying \( \frac{\csc

For each expression, choose the expression below that completes it. Problems involve trigonometric identities and require filling in the blanks with the correct expressions from a list of choices.

Solve trigonometric identity problems by selecting the correct expressions to complete each equation. These exercises cover concepts such as Pythagorean identities and trigonometric simplifications, aimed at high school students in Grades 10-12.

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