proving identities....

**Oasis1993** · December 2nd 2009, 07:49 AM

Hey.

a) By using the subsitution t= tan 1/2 x
prove that cosecx- cotx= tan 1/2x

b) use this result to show that tan15= 2- sqrt 3

THANK YOU!

**Grandad** · December 2nd 2009, 10:21 PM

Hello Oasis1993

Originally Posted by Oasis1993

Hey.

a) By using the subsitution t= tan 1/2 x
prove that cosecx- cotx= tan 1/2x

b) use this result to show that tan15= 2- sqrt 3

THANK YOU!

If $t =\tan\tfrac12x$ , then

$\sin x = \frac{2t}{1-t^2}$ and $\tan x = \frac{2t}{1+t^2}$ .

So:

$\csc x -\cot x=\frac{1+t^2}{2t}-\frac{1-t^2}{2t}$

$=\frac{1+t^2-(1-t^2)}{2t}$

$=\frac{2t^2}{2t}$

$=t$

$=\tan\tfrac12x$

Put $x = 15^o$ , noting that $\tan30^o=\frac{1}{\sqrt3}$ and $\sin30^o=\frac12$ , and you're done.

Grandad

Thread: proving identities....