O Level Additional Math - How To Prove Trigonometry Identity

Dear Ms Ong,
i had just taken my class test on simple trigonometric identities and well as further trignonometric identities yesterday.
sadly to say, i left all the proving of trigonometric identities blank which is worth a total of 9 marks.
Im starting to get panic about trigonometric identities for my coming common test,end of year exams as well as the big O Level.
so im actually here to ask you if you may guide me on how to do them in a faster way, because i realised ive no time to tackle those proving questions.
Best Wishes,
Priscilla Fong

I received this email among the many others in my mailbox. I think this is a common problem face in proving Trigo Identity question. I shall provide a few strategies to handle this sort of question.

  • Usually start with the left hand side or whichever side which is more complicated.
  • When given trigo functions in the form of sec, cosec or cot x , it is advisable to change it to basic trigo functions like sin, cos, tan
  • When question involves power (especially square power), formulas should be considered.
  • Have the END in mind before you begin i.e if the end result involves only sin, your working should eventually eliminate the other trigo functions and stick on to sin.
  • It should take less than 8 steps for a 3-4 marks proving question.
  • [pmath]a^2-b^2=(a+b)(a-b)[/pmath] is sometimes used in proving Trigo Identity. (Algebra rule)

I am going to use an example to illustrate how to use the above strategies

Q: Prove the identity [pmath](sec A - tan A)^2 = {1- sin A}/{1+ sin A}[/pmath]

[pmath](sec A - tan A)^2 =( {1}/{ cos A}-{sin A}/{cos A})^2[/pmath] (Change sec and tan into sin & cos)

=  [pmath]( {1 - sin A}/{cos A})^2[/pmath]

= [pmath]{(1-sin A)^2}/{(cos A)^2} [/pmath]

= [pmath] {(1 - sin A)(1 - sin A)}/{(1 - sin^2 A)} [/pmath] (applying formula)

= [pmath] {(1 - sin A)(1 - sin A)}/{(1 - sin A)(1 + sin A)} [/pmath] (applying algebraic rule)

= [pmath]{1 - sin A}/{1 + sin A} [/pmath] (Proven)

Total number of steps = 6

Additional resource: You may want to read on A-Math Trigo Graphs here.

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Hi, I'm Ai Ling Ong. I enjoy coaching students who have challenges with understanding and scoring in 'O' Level A-Maths and E-Maths. I develop Math strategies, sometimes ridiculous ideas to help students in understanding abstract concepts the fast and memorable way. I write this blog to share with you the stuff I teach in my class, the common mistakes my students made, the 'way' to think, analyze... If you have found this blog post useful, please share it with your friends. I will really appreciate it! :)

6 Responses to O Level Additional Math - How To Prove Trigonometry Identity

  1. how do i master proving of trig identities with ease...i will be glad if i can get a summary note on the proving and tips in proving from u


  2. This example is wrong and makes no sense. (1-sin)^2 equals (1-sin)(1+sin)


    Ai Ling Reply:

    (1-sinx)^2 = (1-sinx)(1-sinx)
    1-sin^2x = (1-sinx)(1+sinx)
    Note: sin^2x is read as sin square x


    Everything is correct in the example.


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