Miss Loi avatar

Trigonometry – You’re An A1 Student If You Can Prove This!

Tuition given in the topic of A-Maths Tuition Questions from the desk of Miss Loi at 5:00 pm (Singapore time)

Updated on

In case you’re wondering, the previous trigonometry proving question here was just a warm-up. This then, guys and gals, is the real deal. If you can prove this, you are A1/A-List/Elite (or whatever honour our society decides to bestow upon you) student material and Miss Loi will kneel down and kow-tow to you!

Just joking lah – Miss Loi’s poor tender forehead will become blue-black if this happens!

Prove the following trigonometric identity:

    \[ \frac{\tan x+\sec x-1}{\tan x-\sec x+1} \equiv \sec x+\tan x \]

P.S. If you’re (predictably) stuck, think of how you rationalize your surds. If it’s of any comfort to you, Miss Loi was stumped too when she first saw this 😛


Revision Exercise

To show that you have understood what Miss Loi just taught you from the centre, you must:

  1. Leave A Comment!
Comments & Reactions


  1. Ms. S@ddist's Avatar
    Ms. S@ddist commented in tuition class


    Ms S@ddist was looking for some trigo identity qn for her revision... and tadaa~
    the holy "杀"人不眨眼 cum 毒辣(crude) cum 欠打的 Ms S@ddist is here!

    Interesting Qn..
    definitely NOT a qn tt can dtm if the one who ans correctly is an A student or not..

    Ms S@ddist shall prove how a Need-Add-Tons-of-Oil student can do this qn as well..

    Quote:"If you can prove this, you are A1/A-List/Elite....Miss Loi will kneel down and kow-tow to you!"


    拜托.. no need for such over exaggerate reaction de pls...
    tis qn is interesting..
    but it's still no-kick for it can be gao dim in 50 SECs..

    A Trigo-identity Pro will ans this way..
    (*note that even though u may be a pro in this field, it doesnt ensure ur A still.. so WAKE up and work on ur other weak areas too..)

    From L.H.S

    {tan x + sec x - 1}/ {tan x-sec x + 1}
    ={(tan x + sec x-1)/(tan x -sec x +1)} * {(tan x - sec x - 1)/(tan x - sec x - 1)}
    = {(tan^2 x - 2tan x - sec^2 x  + 1)/ (tan^2 x - 2tan x sec x - 1 + sec^2 x)}
    =R.H.S (proved)

    See... ahh~ simple rite?? 50 SECs only

    Now lets see how a 差不多 and TTM( think too much) student ans tis qn..
    (*note.. Ms S@ddist suggest that if ur mental calculation(MC) quite telok, do it on a piece of paper itself if u got lost in the following steps..the working will be longer w/o MC for those are redundant steps..
    nah nah... dun tell Ms S@ddist tt it's for clearer presentation and for better revision later on.. if u still need such redundant steps for REVISION, tt only show how "much" u have improved..)

    From L.H.S
    {tan x + sec x - 1}/ {tan x-sec x + 1}
    ={(tan x + sec x-1)/(tan x -sec x +1)} * {(tan x - sec x - 1)/(tan x - sec x - 1)}
    ={(tan^2 x - 2tan x - sec^2 x + 1)/ (tan^2 x - 2tan x sec x + sec^2 x - 1)}
    ={(-2tan x)/ ( 2 tan^2 - 2 tan x sec x)}
    ={ - 1/(tan x - sec x )}
    yawn~rationalise AGAIN.....
    ={ - (tan x + sec x)/( tan^2 x - sec^2 x)}
    = tan x - sec x
    =R.H.S (proved)

    no need to make ur life so complicated actually... by rationalising doesnt show tt u're smart anyway..
    for math is still about SPEED and ACCURACY..
    e clock still ticks when u are flaunting away u know?

    A NATO student will approach this way by using these identities

    tan x = {sin x} / {cos x} and.. sec x = 1 / {cos x}

    Ms S@ddist not so jek ak to put u guys to sleep w tt one big chunk of workings..

    This Qn probably only worth 3 marks or so according to the standard Ms S@ddist guess..

    So no need waste time to ponder n ponder and to make it worst, still stuck.

    dun sad dun sad..
    add oil not enough then burn oil..

    Pardon for Ms S@ddist's crudeness for tt's how she live up for her name..
    she's still a teenager anyway...

    ah mi tou fo...
    have mercy on this crude little Ms S@ddist 吧..

    O my 天...
    Ms S@ddist din notice the date of this post..
    aye aye..
    wonder will anyone read Ms S@ddist's comment..

    “ Only those who do nothing make no mistakes”---Ms S@ddist.

  2. Miss Loi's Avatar
    Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class


    After sneaking in here dressed in a black 夜行衣 to leave her workings on a 9-month-old post, this overconfident 杀手 Ms Saddist inadvertently set off a 机关 that alerted Miss Loi to her presence.

    After examining her 50 SEC workings, she is of the opinion that examiners, at least those who have not heard of this 传说人物, will typically expect most students to be of the 差不多 and TTM kind. As such they will tend to think that the student does not know their sec2x = tan2x + 1 and end up being quite jek ak by MINUSing her marks for incomplete working accordingly.

    Well guess it’s time to think of something more interesting. Time to also upgrade her rusty 机关 to prepare for more arrivals like this.

  3. H2SO4's Avatar
    H2SO4 commented in tuition class


    omg. i went to expand the tanx -> sin x/ cos x.
    and blah blah blah blah. >.< OMG OMG OMG. and the best thing was, i didnt get an answer.

  4. sam's Avatar
    sam commented in tuition class


    can u plz help me prove this?
    tanA / 1 - cot A + cot A /1-tan A = sec A.cosec A+1

  5. whatamidoingwithlife's Avatar
    whatamidoingwithlife commented in tuition class


    Start from RHS multiply numerator and denominator by tan x - sec x + 1. Expand the numerator (sec x + tan x)(tan x - sec x+1) and identify that (tan^2x-sec^2x)=-1 and you will prove this identity within 30 seconds or so. Sorry for the reply to the really old post — I was stuck with PSLE math at that point in time and I really think this question is a good reminder that there are multiple solutions. The complicated side may not be the easier side and a tinge of creativity is needed. Laziness for the win!
    ~Some kid with no life <3

  6. Patrick's Avatar
    Patrick commented in tuition class


Post a Comment

  • * Required field
  • Your email will never, ever be published nor shared with a Nigerian banker or a pharmaceutical company.
  • Spam filter in operation.
    DO NOT re-submit if your comment doesn't appear.
  • Spammers often suffer terrible fates.

Impress Miss Loi with beautiful mathematical equations in your comment by enclosing them within alluring \LaTeX [tex][/tex] tags (syntax guide | online editor), or the older [pmath][/pmath] tags (syntax guide). Please PREVIEW your equations before posting!


Whatsapp Instagram Twitter Facebook Close Search Login Access RSS Joss Sticks Sessions Suggested Solutions Preview O Level Additional Mathematics O Level Elementary Mathematics Secondary Three Additional Mathematics Secondary Three Elementary Mathematics Secondary Two Mathematics Secondary One Mathematics Additional Mathematics 4038 Additional Mathematics 4018 Elementary Mathematics 4017 Virus Zoom Date Modified Address Telephone 非常に人気の Popular Slot! Cart Exam Paper Cart