Verify the identity: `1/(tan^2x) - 1/(cot^2x) = csc^2x - sec^2x`

Expert Answers

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

We have to prove the identity:

`1/(tan^2x) - 1/(cot^2x) = csc^2x - sec^2x`

`1/(tan^2x) - 1/(cot^2x)`

=> `(cos^2x)/(sin^2x) - (sin^2x)/(cos^2x)`

=> `(1 - sin^2x)/(sin^2x) - (1 - cos^2x)/(cos^2x)`

=> `1/(sin^2x) - 1 - 1/(cos^2x) + 1`

=> `1/sin^2x - 1/cos^2x`

=> `csc^2x - sec^2x`

This proves the identity: `1/(tan^2x) - 1/(cot^2x) = csc^2x - sec^2x`

Approved by eNotes Editorial Team

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