Prove the identity. `(1-cosA+sinA)/(1-cosA)=(1+sinA+cosA)/(sinA)`

Expert Answers

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


To prove, let's try to simplify left side. To do so, multiply its numerator and denominator  by the conjugate of 1 - cosA.


`(1-cosA+sinA -cosA + cos^2A + sinAcosA)/(1+cosA-cosA - cos^2A)=(1+sinA+cosA)/(sinA)`

`(1 - cos^2A + sinA+ sinAcosA)/(1-cos^2A)=(1+sinA+cosA)/(sinA)`

Then, apply the Pythagorean identity. So `1 - cos^2A = sin^2A` .

`(sin^2A + sinA + sinAcosA)/(sin^2A)=(1+sinA+cosA)/(sinA)`

Factor out the GCF in the numerator.

`(sinA (sinA + 1 +cosA))/(sin^2A)=(1+sinA+cosA)/(sinA)`

Cancel the common factor between the numerator and denominator.

`(sinA + 1 + cosA)/(sinA)=(1+sinA+cosA)/(sinA)`

Then re-writ the left side as:

`(1+ sinA+cosA)/(sinA)=(1+sinA+cosA)/(sinA) (True)`

The simplified form of the left side is the same with the right side. Hence, this proves that the given equation is an identity.

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