In​ 1-8, match each expression with the correct choice to form an identity. Choices may be used​ once, more than​ once, or not at all.
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Part 1
Drag each of the expressions given above into the appropriate area below.
1. cosine left parenthesis x plus y right parenthesisequals
2. cosine left parenthesis x minus y right parenthesisequals
3. cosine left parenthesis StartFraction pi Over 2 EndFraction minus x right parenthesis
equals
4. sine left parenthesis StartFraction pi Over 2 EndFraction minus x right parenthesis
equals
5. cosine left parenthesis x minus StartFraction pi Over 2 EndFraction right parenthesis
equals
6. sine left parenthesis x minus StartFraction pi Over 2 EndFraction right parenthesis
equals
7. tangent left parenthesis StartFraction pi Over 2 EndFraction minus x right parenthesis
equals
8. cotangent left parenthesis StartFraction pi Over 2 EndFraction minus x right parenthesis
equals
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To match each of these trigonometric expressions with their correct identities, let’s review the identities associated with angle sum and difference formulas and co-function identities:

### Sum and Difference Identities:
1. **cos(x + y)** = cos(x)cos(y) - sin(x)sin(y)
2. **cos(x - y)** = cos(x)cos(y) + sin(x)sin(y)

### Co-function Identities (for angles involving \( \frac{\pi}{2} \)):
3. **cos(π/2 - x)** = sin(x)
4. **sin(π/2 - x)** = cos(x)
5. **cos(x - π/2)** = -sin(x)
6. **sin(x - π/2)** = -cos(x)
7. **tan(π/2 - x)** = cot(x)
8. **cot(π/2 - x)** = tan(x)

Using these identities, here are the matches:

1. **cos(x + y)** = cos(x)cos(y) - sin(x)sin(y)
2. **cos(x - y)** = cos(x)cos(y) + sin(x)sin(y)
3. **cos(π/2 - x)** = sin(x)
4. **sin(π/2 - x)** = cos(x)
5. **cos(x - π/2)** = -sin(x)
6. **sin(x - π/2)** = -cos(x)
7. **tan(π/2 - x)** = cot(x)
8. **cot(π/2 - x)** = tan(x)

Would you like a further breakdown or explanation of each identity? 

Here are some practice questions to further explore these concepts:

1. How does the co-function identity for sine relate to the identity for cosine?
2. What would be the result of cos(π - x) in terms of cos(x)?
3. How can the identity for tan(x + y) be derived using sine and cosine?
4. If sin(x) = 3/5 and x is in the first quadrant, what is cos(π/2 - x)?
5. What is the identity for sec(π/2 - x)?

**Tip:** Remember that co-function identities are particularly useful when you encounter angles involving \( \frac{\pi}{2} \) or 90 degrees.

In 1-8, match each expression with the correct choice to form an identity. Choices may be used once, more than once, or not at all.

1. cosine (x + y) = ?
2. cosine (x - y) = ?
3. cosine (π/2 - x) = ?
4. sine (π/2 - x) = ?
5. cosine (x - π/2) = ?
6. sine (x - π/2) = ?
7. tangent (π/2 - x) = ?
8. cotangent (π/2 - x) = ?

Explore how to match trigonometric identities involving angle sum and difference formulas as well as co-function identities. This detailed guide covers essential trigonometric expressions like cos(x + y), sin(π/2 - x), and cotangent identities. Perfect for high school students in Grades 10-12.

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