Proving Trigonometric Identities Prove: sin2x/1 + cos2x = tanx

Expert Answers

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

We have to prove that: sin 2x/ (1 + cos 2x) = tan x

To do this we use the relations : sin 2x = 2 sin x*cos x and cos 2x = (cos x)^2 - (sin x)^2.

sin 2x/ (1 + cos 2x)

=> [2 sin x*cos x] / [1 + (cos x)^2 - (sin x)^2]

=> [2 sin x*cos x] / [1  - (sin x)^2 + (cos x)^2]

=> [2 sin x*cos x] / [2* (cos x)^2]

=> sin x / cos x

=> tan x

Therefore sin 2x/ (1 + cos 2x) = tan x.

 

Approved by eNotes Editorial Team
An illustration of the letter 'A' in a speech bubbles

sin2x / (1+cos2x) = tanx

We will use trigonometric identities to solve.

We will start from the left side and prove the right side.

==> we know that:

sin2x - 2sinx*cosx

cos2x = 2cos^2 x -1

We will substitute.

==> sin2x / (1+ cos2x) = 2sinx*cosx / (1+ 2cos^2 x -1)

                                      = 2sinx*cosx/ 2cos^2 x

We will reduce similar.

==> sin2x / (1+ cos2x) = sinx/cosx

But we know that tanx = sinx/cosx

==> sin2x / (1+ cos2x) = tanx...........q.e.d

 

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