proving expressions.

**Tweety** · July 31st 2009, 12:29 PM

Given that $a = 4 sec x , b = cosx , c = cotx$

a) express b in terms of a
b) show that $c^{2} = \frac{16}{a^{2} - 16}$

a) $b = cosx$
$a = \frac{4}{cosx}$
$a = \frac{4}{b}$
$b = \frac{4}{a}$

having trouble with part 'b';

**skeeter** · July 31st 2009, 12:33 PM

Originally Posted by Tweety

a) $b = cosx$
$a = \frac{4}{cosx}$
$a = \frac{4}{b}$
$b = \frac{4}{a}$

having trouble with part 'b';

$\frac{16}{(4\sec{x})^2 - 16}$

$\frac{16}{16\sec^2{x} - 16}$

$\frac{1}{\sec^2{x} - 1}$

$\frac{1}{\tan^2{x}}$

$\cot^2{x} = c^2$

**Tweety** · July 31st 2009, 12:54 PM

Originally Posted by skeeter

$\frac{16}{(4\sec{x})^2 - 16}$

$\frac{16}{16\sec^2{x} - 16}$

$\frac{1}{\sec^2{x} - 1}$

$\frac{1}{\tan^2{x}}$

$\cot^2{x} = c^2$

I dont really understand what you did, aren't you meant to prove that $c^{2} = \frac{16}{a^{2}-16}$ ? Why are you doing $c^{2} = cot^2{x}$ ?

This is what I have done but it does not fall together!

$c = cotx , c^{2} = cot^{2}x$

$b = \frac{4}{a} , b^{2} = \frac{16}{a^{2}}$

$cos^{2}x = \frac{16}{a^{2}}$

$1-sin^{2}x = \frac{16}{a^{2}}$

$a^{2}(1-sin^{2}x) = 16$

$a^{2} = \frac{16}{1-sin^{2}x}$

$a^{2} = \frac{16}{1} - \frac{16}{sin^{2}x}$

$a^{2} = 16 - 16cosec^2x$

$a^2 = 16 - 16(1+cot^{2}x)$

$a^{2}= 16- 16(1+c^{2})$

$a^{2} -16 =-16( 1+c^{2})$

is this right so far? But I am stuck here,

**skeeter** · July 31st 2009, 01:00 PM

Originally Posted by Tweety

I dont really understand what you did, aren't you meant to prove that $c^{2} = \frac{16}{a^{2}-16}$ ? Why are you doing $c^{2} = cot^2{x}$ ?

This is what I have done but it does not fall together!

$c = cotx , c^{2} = cot^{2}x$

$b = \frac{4}{a} , b^{2} = \frac{16}{a^{2}}$

$cos^{2}x = \frac{16}{a^{2}}$

$1-sin^{2}x = \frac{16}{a^{2}}$

$a^{2}(1-sin^{2}x) = 16$

$a^{2} = \frac{16}{1-sin^{2}x}$

$a^{2} = \frac{16}{1} - \frac{16}{sin^{2}x}$ this step is incorrect ... you cannot "split" the denominator like that.

...

**Tweety** · July 31st 2009, 01:07 PM

Originally Posted by skeeter

...

Okay, so how would I 'carry' on from there?

Thanks

**AlephZero** · July 31st 2009, 01:34 PM

Originally Posted by Tweety

Okay, so how would I 'carry' on from there?

Thanks

$\sin^2 x + \cos^2 x =1,$ so $1-\sin^2 x =$ ...

**skeeter** · July 31st 2009, 02:17 PM

Originally Posted by Tweety

I dont really understand what you did, aren't you meant to prove that $c^{2} = \frac{16}{a^{2}-16}$ ? Why are you doing $c^{2} = cot^2{x}$ ?

I started with $\frac{16}{a^2-16}$ and showed it was equal to $c^2$ .

note that one may start on either side of an equation and work to get the other side.

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