Let's match each expression with its equivalent from the options.

1. $\sin^2 u$

• Using the Pythagorean identity: $\sin^2 u = 1 - \cos^2 u$.
• Match: $d. \, 1 - \cos^2 u$.

2. $\csc^2 u$

• Using the identity: $\csc^2 u = 1 + \cot^2 u$.
• Match: $a. \, 1 + \cot^2 u$.

3. $\cos \left( \frac{\pi}{2} - u \right)$

• Using the co-function identity: $\cos \left( \frac{\pi}{2} - u \right) = \sin u$.
• Match: $g. \, \sin u$.

4. $\sec^2 u$

• Using the identity: $\sec^2 u = 1 + \tan^2 u$.
• Match: $c. \, 1 + \tan^2 u$.

5. $\csc \left( \frac{\pi}{2} - u \right)$

• Using the co-function identity: $\csc \left( \frac{\pi}{2} - u \right) = \sec u$.
• Match: $b. \, \sec u$.

6. $\tan(-u)$

• Using the odd function property: $\tan(-u) = -\tan(u)$.
• Match: $i. \, -\tan(u)$.

7. $\cos(-u)$

• Using the even function property: $\cos(-u) = \cos(u)$.
• Match: $e. \, \cos u$.

Final Matches

Here is the list of expressions and their equivalents:

1. $\sin^2 u$ → $d. \, 1 - \cos^2 u$
2. $\csc^2 u$ → $a. \, 1 + \cot^2 u$
3. $\cos \left( \frac{\pi}{2} - u \right)$ → $g. \, \sin u$
4. $\sec^2 u$ → $c. \, 1 + \tan^2 u$
5. $\csc \left( \frac{\pi}{2} - u \right)$ → $b. \, \sec u$
6. $\tan(-u)$ → $i. \, -\tan(u)$
7. $\cos(-u)$ → $e. \, \cos u$

Would you like a detailed explanation for any of these matches?

Related Questions

1. What are other examples of trigonometric identities?
2. How do co-function identities work in trigonometry?
3. Can you explain the difference between even and odd trigonometric functions?
4. How are Pythagorean identities derived in trigonometry?
5. What are the applications of trigonometric identities in calculus?

Tip

To remember co-function identities, note that functions like sine and cosine, tangent and cotangent, secant and cosecant are paired such that each function equals its co-function at complementary angles.