6.4: Double, Half, and Power Reducing Identities
( \newcommand{\kernel}{\mathrm{null}\,}\)
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=cos2x−sin2x
- tan2x=2tanx1−tan2x
Half Angle Identities
The half angle identities are a rewritten version of the power reducing identities. The proofs are left as review problems.
- sinx2=±√1−cosx2
- cosx2=±√1+cosx2
- tanx2=±√1−cosx1+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=1−cos2x2
- cos2x=1+cos2x2
- tan2x=1−cos2x1+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 sinx⋅sinx⋅sinx⋅sinx does technically simplify this expression as necessary, you should try to get the terms to sum together not multiply together.
sin4x=(sin2x)2=(1−cos2x2)2=1−2cos2x+cos22x4=14(1−2cos2x+1+cos4x2)
Examples
Earlier, you were asked to find sin215∘. In order to fully identify sin215∘ you need to use the power reducing formula.
sin2x=1−cos2x2sin215∘=1−cos30∘2=12−√34=2−√34
Write the following expression with only sinx and cosx:sin2x+cos3x.
sin2x+cos3x=2sinxcosx+cos(2x+x)=2sinxcosx+cos2xcosx−sin2xsinx=2sinxcosx+(cos2x−sin2x)cosx−(2sinxcosx)sinx=2sinxcosx+cos3x−sin2xcosx−2sin2xcosx=2sinxcosx+cos3x−3sin2xcosx
Use half angles to find an exact value of tan 22.5∘ without using a calculator.
tanx2=±√1−cosx1+cosxtan22.5∘=tan45∘2=±√1−cos45∘1+cos45∘=±√1−√221+√22=±√22−√2222+√22=±√2−√22+√2=±√(2−√2)22
Prove the power reducing identity for sine.
sin2x=1−cos2x2
Using the double angle identity for cosine:
cos2x=cos2x−sin2xcos2x=(1−sin2x)−sin2xcos2x=1−2sin2x
This expression is an equivalent expression to the double angle identity and is often considered an alternate form.
Simplify the following identity. sin4x−cos4x.
Here are the steps:
sin4x−cos4x=(sin2x−cos2x)(sin2x+cos2x)=−(cos2x−sin2x)=−cos2x
Review
Prove the following identities.
1. sin2x=2sinxcosx
2. cos2x=cos2x−sin2x
3. tan2x=2tanx1−tan2x
4. cos2x=1+cos2x2
5. tan2x=1−cos2x1+cos2x
6. sinx2=±√1−cosx2
7. cosx2=±√1+cosx2
8. tanx2=±√1−cosx1+cosx
9. csc2x=12cscxsecx
10. cot2x=cot2x−12cotx
Find the value of each expression using half angle identities.
11. tan15∘
12. tan22.5∘
13. sec22.5∘
14. Show that tanx2=1−cosxsinx
15. Using your knowledge from the answer to question 14, show that tanx2=sinx1+cosx.