1-cosx/2 / 1+cosx/2
- print Print
- list Cite
Expert Answers
calendarEducator since 2011
write5,349 answers
starTop subjects are Math, Science, and Business
Supposing that you need to simplify the expression `((1-cos x)/2))/((1 + cos x)/2))` , you should use the half angle formulas such that:
`sin(x/2) = sqrt((1-cos x)/2) => sin^2(x/2) = (1-cos x)/2`
`cos(x/2) = sqrt((1+cos x)/2) => cos^2(x/2) = (1+cos x)/2`
Hence, substituting `sin^2(x/2)` for `(1-cos x)/2` and `(1+cos x)/2` for `cos^2(x/2)` yields:
`((1-cos x)/2)/((1+cos x)/2) = (sin^2(x/2))/(cos^2(x/2))`
You need to remember that sin alpha/cos alpha = tan alpha, hence, reasoning by analogy yields:
`(sin^2(x/2))/(cos^2(x/2)) = (sin(x/2))/(cos(x/2))*(sin(x/2))/(cos(x/2)) = tan(x/2)*tan(x/2) = tan^2 (x/2)`
Hence, simplifying the given expression yields `((1-cos x)/2)/((1+cos x)/2) = tan^2 (x/2).`
Related Questions
- Prove that `cos (x/2)=sqrt ((1/2)(1 + cosx))`
- 1 Educator Answer
- Prove the identity sinx/2=squareroot(1-cosx)/2.
- 1 Educator Answer
- Prove ((sinx+cosx)/(sinx-cosx))+((sinx-cosx)/(sinx+cosx)) = (2sec^2 x/tan^2 x)-1
- 1 Educator Answer
- Verify the identity sinx/(1-cosx)=(1+cosx)/sinx.
- 1 Educator Answer
- Verify: (cosx/1 + sinx) + (cosx/1-sinx) = 2secx
- 1 Educator Answer
Unlock This Answer Now
Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.