Hello, bigroo!
$\displaystyle \text{I know from my conic that: }\:\tan2\theta\:=\:\dfrac{1}{4}$
$\displaystyle \text{Can you show me how I arrive at: }\:\cos2\theta\:=\:\dfrac{4}{\sqrt17}}$
We have: .$\displaystyle \tan2\theta \:=\:\dfrac{1}{4} \:=\:\dfrac{opp}{adj}$
$\displaystyle \,2\theta$ is in a right triangle with: .$\displaystyle opp = 1,\;adj = 4$
Code:
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| * __
| * √17
1 | *
| *
| 2@ *
* - - - - - - - - - - - *
4
Pythagorus says: .$\displaystyle hyp^2 \:=\:1^2 + 4^2 \:=\:17 \quad\Rightarrow\quad hyp \:=\:\sqrt{17}$
Therefore: .$\displaystyle \cos2\theta \;=\;\dfrac{adj}{hyp} \;=\;\dfrac{4}{\sqrt{17}} $