prove the identity: 2cos^2 theta/2 = sin^2 theta/1-cos theta.
- print Print
- list Cite
Expert Answers
calendarEducator since 2015
write85 answers
starTop subject is Math
Prove
`2cos^2(theta/2) = (sin^2theta)/(1-costheta)`
The half angle identity states:
`cos(theta/2) = sqrt((1+costheta)/2)`
This turns out left side into,
`2(1+costheta)/2 =(sin^2theta)/(1-costheta)`
`1+costheta =(sin^2theta)/(1-costheta)`
`(1-costheta)/(1-costheta) (1+costheta) =(sin^2theta)/(1-costheta)`
This simplifies into
`(1-cos^2theta)/(1-costheta) =(sin^2theta)/(1-costheta)`
Which turns into
`(sin^2theta)/(1-costheta) =(sin^2theta)/(1-costheta)`
It is proven!
Related Questions
- prove that sin^4(theta)-cos^4(theta)=sin^2(theta)-cos^2(theta)trigonometry
- 1 Educator Answer
- How to prove the identity `sin^2x + cos^2x = 1` ?
- 3 Educator Answers
- Prove the identity sin x - sin y = 2 sin( (x - y)/2 ) cos( (x + y)/2 )?
- 2 Educator Answers
- Prove: `cos^4 x + 1 - sin^4 x = 2cos^2 x`
- 1 Educator Answer
- Prove the identity: (cos x + cos y)^2 + (sin x – sin y)^2 = 2 + 2cos(x + y)
- 1 Educator Answer
briefcaseCollege Professor
bookPh.D. from Oregon State University
calendarEducator since 2015
write3,395 answers
starTop subjects are Science, Math, and Business
Using the following identities:
`sin^2 x + cos^2 x =1`
and `cos^2 x = (1+cos 2x)/2`
Left hand side = 2 cos^2 (theta/2) = 2. (1+cos 2.theta/2)/2 = 1+cos theta
multiplying and dividing by (1-cos theta), we get.
LHS = (1+cos theta). (1-cos theta)/ (1-cos theta) = (1-cos^2 theta)/(1-cos theta)
= (sin^2 theta + cos^2 theta - cos theta)/(1cos theta) = sin^2 theta/ (1-cos theta). = RHS
Hence proved.
using half angle identities
Student Answers