proof

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August 10th 2009, 10:39 AM
Tweety

proof

Given that $x = sec\theta + tan\theta$

show that $\frac{1}{x} = sec\theta - tan\theta$
August 10th 2009, 10:44 AM
red_dog

$\frac{1}{x}=\frac{1}{\displaystyle\frac{1}{\cos\th eta}+\frac{\sin\theta}{\cos\theta}}=\frac{\cos\the ta}{1+\sin\theta}=$

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

$=\frac{1}{\cos\theta}-\frac{\sin\theta}{\cos\theta}=\sec\theta-\tan\theta$
August 10th 2009, 10:55 AM
Tweety

Just been thinking, is this method correct?

$x = sec\theta + tan\theta$

$\frac{1}{x} = sec\theta - tan\theta$

$\frac{1}{sec\theta+tan\theta} = sec\theta - tan\theta$

multiply both sides by $sec\theta+tan\theta$

$1 = ( sec\theta - tan\theta) (sec\theta+tan\theta )$

$\frac{1}{sec\theta - tan\theta} = x$

$1 = x(sec\theta - tan\theta)$

$\frac{1}{x} = sec\theta - tan\theta$
August 10th 2009, 10:56 AM
Bruno J.

This exact question was posted yesterday here : http://www.mathhelpforum.com/math-he...-identity.html
August 10th 2009, 11:00 AM
Bruno J.

Quote:

Originally Posted by Tweety

Just been thinking, is this method correct?

$x = sec\theta + tan\theta$

$\frac{1}{x} = sec\theta - tan\theta$

$\frac{1}{sec\theta+tan\theta} = sec\theta - tan\theta$

multiply both sides by $sec\theta+tan\theta$

$1 = ( sec\theta - tan\theta) (sec\theta+tan\theta )$

$\frac{1}{sec\theta - tan\theta} = x$

$1 = x(sec\theta - tan\theta)$

$\frac{1}{x} = sec\theta - tan\theta$

You seem to be assuming that $ \frac{1}{x} = \sec\theta - \tan\theta $ and that $ {x} = \sec\theta + \tan\theta $ , doing a few operations on these premises, undoing those operations, and then concluding that $ \frac{1}{x} = \sec\theta - \tan\theta $ .
August 10th 2009, 11:02 AM
Tweety

Quote:

Originally Posted by Bruno J.

This exact question was posted yesterday here : http://www.mathhelpforum.com/math-he...-identity.html

Well than they must be using the same excercise book as me, as my exam board is with edexcel, this is the book I am using!

Edexcel AS and A Level Modular Mathematics - Core Mathematics 3: Amazon.co.uk: Mr Keith Pledger: Books
August 10th 2009, 11:23 AM
Tweety

Quote:

Originally Posted by Bruno J.

You seem to be assuming that $ \frac{1}{x} = \sec\theta - \tan\theta $ and that $ {x} = \sec\theta + \tan\theta $ , doing a few operations on these premises, undoing those operations, and then concluding that $ \frac{1}{x} = \sec\theta - \tan\theta $ .

Well yes becasue thats what the book says?

can you tell me what steps are not 'valid'?

Also when I have the expression $1 = ( sec\theta - tan\theta) (sec\theta+tan\theta )$ and than divide most sides by $sec\theta - tan\theta$

would that not cancel $sec\theta - tan\theta$ from the right hand side? so I get this expression; $\frac{1}{sec\theta - tan\theta} = sec\theta+tan\theta$

thanks.
August 10th 2009, 02:29 PM
Bruno J.

No, you are given that $ {x} = \sec\theta + \tan\theta $ , from which you must deduce that $ \frac{1}{x} = \sec\theta - \tan\theta $ .

You cannot use in your proof that $ \frac{1}{x} = \sec\theta - \tan\theta $ because that is what you are trying to show. However that is what you seem to be doing in the third line : starting from $ \frac{1}{x} = \sec\theta - \tan\theta $ , you replaced $x$ by $ \sec\theta + \tan\theta $ .
August 11th 2009, 02:20 AM
Tweety

Quote:

Originally Posted by Bruno J.

No, you are given that $ {x} = \sec\theta + \tan\theta $ , from which you must deduce that $ \frac{1}{x} = \sec\theta - \tan\theta $ .

You cannot use in your proof that $ \frac{1}{x} = \sec\theta - \tan\theta $ because that is what you are trying to show. However that is what you seem to be doing in the third line : starting from $ \frac{1}{x} = \sec\theta - \tan\theta $ , you replaced $x$ by $ \sec\theta + \tan\theta $ .

oh I get what you meant now,

Thanks