Prove the identity
(sinx+cosx)2=1+sin2x.
We have
(sinx+cosx)2=sin2x+cos2x+2sinxcosx=1+2sinxcosx=1+sin2x. □
Prove the identity
sin2x=tanx(1+cos2x).
We have
cosxsinx(1+cos2x)=cosxsinx(1+2cos2x−1)=cosxsinx(2cos2x)=2sinxcosx=sin2x. □
Prove the identity
cos2x=cotx+tanxcotx−tanx.
We have
cotx+tanxcotx−tanx=sinxcosx+cosxsinxsinxcosx−cosxsinx=cos2x+sin2xcos2x−sin2x=cos2x. □
Here goes some important terminology:
- versed sine x: vers(x)=1−cosx=2sin22x
- versed cosine x: vercos(x)=1+cosx=2cos22x
- coversed sine x: covers(x)=1−sinx=(sin2x−cos2x)2
- coversed cosine x: covercos(x)=1+sinx=(sin2x+cos2x)2
- chord(x)=crd(x)=2sin2x.