Math Problem Statement

Match each expression with its equivalent: 1. sin^2 u, 2. csc^2 u, 3. cos(π/2 - u), 4. sec^2 u, 5. csc(π/2 - u), 6. tan(-u), 7. cos(-u).

Solution

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

1. sin2u\sin^2 u

  • Using the Pythagorean identity: sin2u=1cos2u\sin^2 u = 1 - \cos^2 u.
  • Match: d.1cos2ud. \, 1 - \cos^2 u.

2. csc2u\csc^2 u

  • Using the identity: csc2u=1+cot2u\csc^2 u = 1 + \cot^2 u.
  • Match: a.1+cot2ua. \, 1 + \cot^2 u.

3. cos(π2u)\cos \left( \frac{\pi}{2} - u \right)

  • Using the co-function identity: cos(π2u)=sinu\cos \left( \frac{\pi}{2} - u \right) = \sin u.
  • Match: g.sinug. \, \sin u.

4. sec2u\sec^2 u

  • Using the identity: sec2u=1+tan2u\sec^2 u = 1 + \tan^2 u.
  • Match: c.1+tan2uc. \, 1 + \tan^2 u.

5. csc(π2u)\csc \left( \frac{\pi}{2} - u \right)

  • Using the co-function identity: csc(π2u)=secu\csc \left( \frac{\pi}{2} - u \right) = \sec u.
  • Match: b.secub. \, \sec u.

6. tan(u)\tan(-u)

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

7. cos(u)\cos(-u)

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

Final Matches

Here is the list of expressions and their equivalents:

  1. sin2u\sin^2 ud.1cos2ud. \, 1 - \cos^2 u
  2. csc2u\csc^2 ua.1+cot2ua. \, 1 + \cot^2 u
  3. cos(π2u)\cos \left( \frac{\pi}{2} - u \right)g.sinug. \, \sin u
  4. sec2u\sec^2 uc.1+tan2uc. \, 1 + \tan^2 u
  5. csc(π2u)\csc \left( \frac{\pi}{2} - u \right)b.secub. \, \sec u
  6. tan(u)\tan(-u)i.tan(u)i. \, -\tan(u)
  7. cos(u)\cos(-u)e.cosue. \, \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.

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Math Problem Analysis

Mathematical Concepts

Trigonometric Identities
Pythagorean Identities
Co-function Identities
Odd and Even Properties

Formulas

sin^2 u = 1 - cos^2 u
csc^2 u = 1 + cot^2 u
cos(π/2 - u) = sin u
sec^2 u = 1 + tan^2 u
csc(π/2 - u) = sec u
tan(-u) = -tan(u)
cos(-u) = cos(u)

Theorems

Pythagorean Identities
Co-function Identities
Odd and Even Properties

Suitable Grade Level

Grades 10-12