1. ## proving identities....

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!

2. 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.