Expert Answers

An illustration of the letter 'A' in a speech bubbles

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).`

Approved by eNotes Editorial Team
Soaring plane image

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial