Here's a really basic intro to some trigonometric identities. If you're struggling with the practice problems, you can refer to this video for some help.


















This is an easy to follow and simple video that covers basic problems to help you learn and further understand trigonometry identities.

Practice Problems & Solutions: To view the answers, simply highlight the space to the right of the bubbles! There are also multiple ways to solve the same problem! Do not be discouraged if you did not solve it in the same manner. As long as all of the steps that you did are mathematically correct, it is still okay as you are ultimately arriving at the same answer. That's the beauty of proofs as you are allowed to think differently :).

  1. sec2A + tan2Asec2A = sec4A
    • sec2A (1+tan2A) = sec4A
    • sec2A * sec2A = sec4A
    • sec4A = sec4A

  2. sec4x - tan4x = 1 + 2tan2x
    • (sec2x + tan2x)(sec2x - tan2x) = 1 + 2tan2x
    • sec2x + tan2x = 1 + 2tan2x
    • 1 + tan2x + tan2x = 1 + 2tan2x
    • 1 + 2tan2x = 1 + 2tan2x

  3. cosx (secx-cosx) = sin2x
    • cosxsecx - cosxcosx = sin2x
    • 1 - cos2x = sin2x
    • sin2x = sin2x

  4. sin3zcos2z = sin3z - sin5z
    • sin3zcos2z = sin3z(1 - sin2z)
    • sin3zcos2z = sin3zcos2z
  5. sec4A(1-Sec4A) - 2tan2A = 1
  • sec4A - sec4Asin4A - 2tan2A = 1
  • sec4A - sin4A/cos4A - 2tan2A = 1
  • sec4A - tan4A - 2tan2A = 1
  • (sec2A)2 - (tan2A)2- 2tan2A = 1
  • (sec2A+tan2A)(sec2A - tan2A) - 2tan2A = 1
  • (sec2A+tan2A)(1) - 2tan2A = 1
  • sec2A + tan2A - 2tan2A = 1
  • sec2A - tan2A = 1
  • 1 = 1
6. sin2x * tan2x + cos2x * cot2x = tan2x + cot2x - 1
  • sin2x * tan2x + cos2x * cot2x = tan2x + cot2x - 1 => Change sin2x into (1 - cos2x) and cos2x into (1 - sin2x)
  • (1 - cos2x) * tan2x + (1 - sin2x) cot2x = tan2x + cot2x - 1 => Distribute
  • tan2x - cos2x * tan2x + cot2x - sin2x * cot2x = tan2x + cot2x - 1 =>Change tan2x into (sin2x / cos2x) and cot2x into (cos2x / sin2x)
  • tan2x - cos2x * sin2x / cos2x + cot2x - sin2x * cos2x / sin2x = tan2x + cot2x - 1 =>multiply
  • tan2x - sin2x + cot2x - cos2x = tan2x + cot2x - 1 => pull out a negative
  • -(-tan2x + sin2x - cot2x + cos2x) = tan2x + cot2x - 1 => Remember sin2x + cos2x = 1
  • -(-tan2x - cot2x + 1) = tan2x + cot2x - 1 => Pull out another negative
  • tan2x + cot2x - 1= tan2x + cot2x -7>1


7.)Verify the identity cos x * tan x = sin x
(highlight for answer!)



  • We start with the left side and transform it into sin x. Use the identity tan x = sin x / cos x in the left side.

    cos x * tan x = cos x * (sin x / cos x) = sin x



8.)Verify the identity cot x * sec x * sin x = 1
(highlight for answer!)


  • Use the identities cot x = cos x / sin x and sec x = 1/ cos x in the left side.

    cot x * sec x * sin x = (cos x / sin x) * (1/ cos x) * sin x

  • Simplify to obtain.

    (cos x / sin x) * (1/ cos x) * sin x = 1


If you need more practice in finding the sum of trigonometic functions or need to relearn a few things, go to http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-rcostheta-alpha-2009-1.pdf.