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

Help With Proving Trig Identities (1 Viewer)

twinklegal19

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SpiralFlex

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twinklegal19

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