In my work they are asking for the exact values of $\displaystyle \frac{\Pi}{8}$ in all forms: Sin, Cos, Tan and recipricols by using double angle formula's. Sin and Cos of $\displaystyle \frac{\Pi}{8}$ make sense but i have a query involving Tan and Cotan.

i have figured out that $\displaystyle sin\frac{\Pi}{8} = \frac{-\sqrt2+2}{2}$

i have also figured out that $\displaystyle cos\frac{\Pi}{8} = \frac{\sqrt2+2}{2}$

How would i go about figuring out $\displaystyle Tan\frac{\Pi}{8}$, Using the respective double angle formula didn't work so now i am stumped. Also i don't know at which point i should switch the numerator and denominator for Recipricol functions

$\displaystyle Cosec(\frac{\Pi}{8}) = \frac{1}{Sin\frac{\Pi}{8}}$

$\displaystyle = \frac{1}{\frac{-\sqrt2+2}{2}} = \frac{2}{-\sqrt2+2} $ and times by the conjugate to get the answer?

any help would greatly boost my confidence