Had a maths problem in my final Year 11 Exam (2 unit) (1 Viewer)

jadenmaccas · Sep 15, 2011

Hey guys. I'm in Year 10 accelerating 2 unit maths and I came across a trigonomic identity which I found almost impossible to prove. After the exam, I heard that you have to 'cross-multiply' to solve it by some students in Year 11. I never came across this before. Isn't it just LHS = RHS? I couldn't prove the identity correctly.

Here is the question: http://latex.codecogs.com/gif.latex?\frac{1}{\csc%20x%20+%20\cot%20x}%20=%20\csc%20x%20-%20\cot%20x

nightweaver066 · Sep 15, 2011

$\frac{1}{cosecx + cotx} = cosecx + cotx$
$LHS: \frac{1}{\frac{1}{sinx} + \frac{cosx}{sinx}}$
Multiplying everything by sin^2x

$= \frac{sin^2x}{sinx + sinxcosx}$
$= \frac{1 - cos^2x}{sinx(1 + cosx)}$
$= \frac{(1 + cosx)(1-cosx)}{sinx(1+cosx)}$
$= \frac{1 - cosx}{sinx}$
$= \frac{1}{sinx} - \frac{cosx}{sinx}$
$= cosecx - cotx = RHS$

kooliskool · Sep 15, 2011

Hi, it's actually pretty simple, no need to do "cross-multiply" or anything, LHS=....=RHS is the correct way, think of the relationship between cosec and cot:

$LHS=\frac{1}{\csc x+ \cot x}$

$=\frac{\csc x- \cot x}{\left (\csc x+ \cot x \right )\left ( \csc x- \cot x \right )}$

$=\frac{\csc x- \cot x}{\csc ^{2}x - \cot^{2} x}$

$=\frac{\csc x- \cot x}{1}$

since $\csc ^{2}x = 1+ \cot^{2} x$

$\therefore LHS=\csc x- \cot x$

$=RHS$

jadenmaccas · Sep 15, 2011

nightweaver066 said:
$\frac{1}{cosecx + cotx} = cosecx + cotx$
$LHS: \frac{1}{\frac{1}{sinx} + \frac{cosx}{sinx}}$
Multiplying everything by sin^2x

$= \frac{sin^2x}{sinx + sinxcosx}$
$= \frac{1 - cos^2x}{sinx(1 + cosx)}$
$= \frac{(1 + cosx)(1-cosx)}{sinx(1+cosx)}$
$= \frac{1 + cosx}{sinx}$
$= \frac{1}{sinx} + \frac{cosx}{sinx}$
$= cosecx + cotx = RHS$

Nightweaver, read the question -.-

nightweaver066 · Sep 15, 2011

Mistake in the working out third last line.. Corrected.

cutemouse · Sep 16, 2011

What school do you go to?

maths lover · Sep 16, 2011

this was in my prelims as well, this is an independent question.

Had a maths problem in my final Year 11 Exam (2 unit) (1 Viewer)

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