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6.4: Double, Half, and Power Reducing Identities

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These identities are significantly more involved and less intuitive than previous identities. By practicing and working with these advanced identities, your toolbox and fluency substituting and proving on your own will increase. Each identity in this concept is named aptly. Double angles work on finding sin80 if you already know sin40. Half angles allow you to find sin15 if you already know sin30. Power reducing identities allow you to find sin215 if you know the sine and cosine of 30

What is sin215?

Double Angle, Half Angle, and Power Reducing Identities

Double Angle Identities

The double angle identities are proved by applying the sum and difference identities. They are left as review problems. These are the double angle identities.

  • sin2x=2sinxcosx
  • cos2x=cos2xsin2x
  • tan2x=2tanx1tan2x

Half Angle Identities

The half angle identities are a rewritten version of the power reducing identities. The proofs are left as review problems.

  • sinx2=±1cosx2
  • cosx2=±1+cosx2
  • tanx2=±1cosx1+cosx

Power Reducing Identities

The power reducing identities allow you to write a trigonometric function that is squared in terms of smaller powers. The proofs are left as examples and review problems.

  • sin2x=1cos2x2
  • cos2x=1+cos2x2
  • tan2x=1cos2x1+cos2x

Power reducing identities are most useful when you are asked to rewrite expressions such as sin4x as an expression without powers greater than one. While sinxsinxsinxsinx does technically simplify this expression as necessary, you should try to get the terms to sum together not multiply together.

sin4x=(sin2x)2=(1cos2x2)2=12cos2x+cos22x4=14(12cos2x+1+cos4x2)

Examples

Example 1

Earlier, you were asked to find sin215. In order to fully identify sin215 you need to use the power reducing formula.

sin2x=1cos2x2sin215=1cos302=1234=234

Example 2

Write the following expression with only sinx and cosx:sin2x+cos3x.

sin2x+cos3x=2sinxcosx+cos(2x+x)=2sinxcosx+cos2xcosxsin2xsinx=2sinxcosx+(cos2xsin2x)cosx(2sinxcosx)sinx=2sinxcosx+cos3xsin2xcosx2sin2xcosx=2sinxcosx+cos3x3sin2xcosx

Example 3

Use half angles to find an exact value of tan 22.5 without using a calculator.

tanx2=±1cosx1+cosxtan22.5=tan452=±1cos451+cos45=±1221+22=±222222+22=±222+2=±(22)22

Example 4

Prove the power reducing identity for sine.

sin2x=1cos2x2

Using the double angle identity for cosine:
cos2x=cos2xsin2xcos2x=(1sin2x)sin2xcos2x=12sin2x

This expression is an equivalent expression to the double angle identity and is often considered an alternate form.

Example 5

Simplify the following identity. sin4xcos4x.

Here are the steps:
sin4xcos4x=(sin2xcos2x)(sin2x+cos2x)=(cos2xsin2x)=cos2x

Review

Prove the following identities.

1. sin2x=2sinxcosx

2. cos2x=cos2xsin2x

3. tan2x=2tanx1tan2x

4. cos2x=1+cos2x2

5. tan2x=1cos2x1+cos2x

6. sinx2=±1cosx2

7. cosx2=±1+cosx2

8. tanx2=±1cosx1+cosx

9. csc2x=12cscxsecx

10. cot2x=cot2x12cotx

Find the value of each expression using half angle identities.

11. tan15

12. tan22.5

13. sec22.5

14. Show that tanx2=1cosxsinx

15. Using your knowledge from the answer to question 14, show that tanx2=sinx1+cosx.


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