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

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

Updated on 25 June 2015

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 😛

頑張って!!!

Miss Loi

trigonometric identities, trigonometry

Revision Exercise

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

Comments & Reactions

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6 Comments

Ms. S@ddist commented in tuition class
日
曜
日
2008
Jan
13
Sun
3:11am
1
well...
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..
BUT
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!"
alamak..
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

=
= ${(tan^2 x - 2tan x - sec^2 x + 1)/ (tan^2 x - 2tan x sec x - 1 + sec^2 x)}$
=R.H.S (proved)
See... ahh~ simple rite?? 50 SECs only
rite??
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

=
= ${(tan^2 x - 2tan x - sec^2 x + 1)/ (tan^2 x - 2tan x sec x + sec^2 x - 1)}$
= ${(-2tan x)/ ( 2 tan^2 - 2 tan x sec x)}$
=
yawn~rationalise AGAIN.....
= ${ - (tan x + sec x)/( tan^2 x - sec^2 x)}$
=
=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?
ahh..lastly..
A NATO student will approach this way by using these identities
and..
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.
Anyway..
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.
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Miss Loi Friend Miss Loi on Facebook @MissLoi commented in tuition class
月
曜
日
2008
Jan
14
Mon
1:04am
2
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.
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Reply Miss Loi
H2SO4 commented in tuition class
木
曜
日
2008
Oct
23
Thu
10:56pm
3
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.
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sam commented in tuition class
日
曜
日
2011
Apr
10
Sun
6:18pm
4
can u plz help me prove this?
tanA / 1 - cot A + cot A /1-tan A = sec A.cosec A+1
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whatamidoingwithlife commented in tuition class
金
曜
日
2015
Apr
24
Fri
10:18pm
5
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
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Patrick commented in tuition class
日
曜
日
2015
May
17
Sun
1:12pm
6
$\begin{align*} \text{LHS} &= \frac{\tan x + \sec x -1}{\tan x - \sec x +1}\\ &= \frac{\tan x + \sec x -[\sec^{2}x - \tan^{2}x]}{\tan x - \sec x +1}\\ &= \frac{(\sec x + \tan x)[1-(\sec x - \tan x)]}{\tan x - \sec x + 1}\\ &= \text{RHS} \end{align*}$
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