For each​ expression, choose the expression below that completes an identity. One or both expressions may need to be rewritten.

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Part 1

Use the choices below to fill in each blank.

StartFraction 1 Over secant squared x EndFraction1sec2x

cotangent xcotx

cosecant squared x minus cotangent squared x plus sine squared xcsc2x−cot2x+sin2x

negative cosine left parenthesis x right parenthesis−cos(x)

StartFraction sine squared x Over cosine squared x EndFractionsin2xcos2x

39. 

negative cotangent x sine x−cotxsinxequals=

40. 

secant squared x minus 1sec2x−1equals=

41. 

StartFraction cosecant x Over secant x EndFractioncscxsecxequals=

42. 

1 plus sine squared x1+sin2xequals=

43. 

cosine squared xcos2xequals=


Let's analyze and match each expression with the correct identity.

### 39. \( -\cot(x)\sin(x) = ? \)

This expression simplifies using trigonometric identities:
\[
-\cot(x)\sin(x) = -\frac{\cos(x)}{\sin(x)} \cdot \sin(x) = -\cos(x)
\]
**Answer:** \( -\cos(x) \)

### 40. \( \sec^2(x) - 1 = ? \)

This is a well-known Pythagorean identity:
\[
\sec^2(x) - 1 = \tan^2(x)
\]
**Answer:** \( \tan^2(x) \)

### 41. \( \frac{\csc(x)}{\sec(x)} = ? \)

Using the definitions of cosecant and secant:
\[
\frac{\csc(x)}{\sec(x)} = \frac{\frac{1}{\sin(x)}}{\frac{1}{\cos(x)}} = \frac{\cos(x)}{\sin(x)} = \cot(x)
\]
**Answer:** \( \cot(x) \)

### 42. \( 1 + \sin^2(x) = ? \)

This is not a standard identity. However, \( 1 + \sin^2(x) \) could remain as is unless further context is given. No simplification exists without more details.

**Answer:** \( 1 + \sin^2(x) \) (no simplification)

### 43. \( \cos^2(x) = ? \)

There is no need to rewrite this, as it is already a standard expression. However, depending on context, it could be part of the Pythagorean identity:
\[
\cos^2(x) + \sin^2(x) = 1
\]
But in isolation, it remains \( \cos^2(x) \).

**Answer:** \( \cos^2(x) \) (no change)

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Do you need any further details or explanations on these steps?

Here are 5 related questions:
1. How do you prove the identity \( \sec^2(x) - 1 = \tan^2(x) \)?
2. What is the relationship between cotangent and cosine?
3. How do you derive \( \frac{\csc(x)}{\sec(x)} = \cot(x) \)?
4. Can \( 1 + \sin^2(x) \) be rewritten under certain conditions?
5. How is the identity \( \cos^2(x) + \sin^2(x) = 1 \) applied in trigonometry?

**Tip:** The Pythagorean identities are essential tools in simplifying trigonometric expressions.

For each expression, choose the expression below that completes an identity. One or both expressions may need to be rewritten.

Explore how to solve trigonometric identity problems by completing expressions like -cot(x)sin(x) = -cos(x) and sec^2(x) - 1 = tan^2(x). Learn key concepts of trigonometric identities and the Pythagorean identity, suitable for Grades 10-12.

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