Help With Proving Trig Identities (1 Viewer)

twinklegal19 · Jun 4, 2012

I'm stuck on a problem in Cambridge Year 11 Chapter 4G Question 16 k)

Prove

$\frac{(\sin ^{2}\alpha -\cos ^2\alpha )(1-\sin \alpha \cos \alpha )}{\cos \alpha (\sec \alpha -\csc\alpha)(\sin ^3\alpha +\cos^3\alpha )}=\sin \alpha$

My answer ended up being

$$LHS$=\sin(1-\sin\alpha\cos\alpha)$

and it took up more than a page :S

Any help is appreciated, thanks!

Carrotsticks · Jun 4, 2012

SpiralFlex · Jun 4, 2012

Note: This is a question I answered for a fellow BOS member via PM

For Greenboxes from SpiralFlex 2011 said:
Hello to you too. I thought you were trying to beat me?

First of all, with these problems, they will never give you something this tedious to expand. So expect a simple trick to solve them without expanding.

$LHS=\frac{(\sin^2 a-\cos^2 a)(1-\sin a \cos a)}{\cos a (\sec a - \csc a)(\sin^3 a+\cos^3a)}$

$=\frac{(\sin^2 a-\cos^2 a)(1-\sin a \cos a)}{\cos a (\sec a - \csc a)(\sin a+\cos a)(\sin^2 a- \sin a \cos a +\cos^2 a)}$

$=\frac{(\sin^2 a-\cos^2 a)(1-\sin a \cos a)}{\cos a (\sec a - \csc a)(\sin a+\cos a)(1- \sin a \cos a)}$

Cancel them out,

$=\frac{(\sin^2 a-\cos^2 a)}{\cos a (\sec a - \csc a)(\sin a+\cos a)}$

Now we have to expand.

$=\frac{\sin^2 a- \cos^2 a}{(1-\frac{\cos a}{\sin a})(\sin a+\cos a)}$

$=\frac{\sin^2 a- \cos^2 a}{\sin a+\cos a - \cos a -\frac{\cos^2 a}{\sin a}}$

$=\frac{\sin^2 a- \cos^2 a}{\sin a -\frac{\cos^2 a}{\sin a}}$

$=\frac{\sin^2 a- \cos^2 a}{\sin a -\frac{\cos^2 a}{\sin a}} * \frac{\sin a}{\sin a}$

$=\frac{\sin a(\sin^2 a- \cos^2 a)}{\sin^2 a - \cos^2 a}$

$=\sin a$

$=RHS$

$\therefore LHS=RHS$

twinklegal19 · Jun 4, 2012

Carrotsticks said:

"You must spread some Reputation around before giving it to Carrotsticks again."

Thank you so much!

I can't believe I didn't see the difference of squares and sum of cubes in the first step before and instead expanded them them all lol.

twinklegal19 · Jun 4, 2012

SpiralFlex said:
Note: This is a question I answered for a fellow BOS member via PM

"You must spread some Reputation around before giving it to SpiralFlex again." hehe

Thank you for your solution! Really appreciated it

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Help With Proving Trig Identities (1 Viewer)

twinklegal19

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Carrotsticks

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SpiralFlex

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twinklegal19

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