Question

How to find tan 22.5 degree with shortest method?

Answer 1

we know $tan{45}^{\circ }=1$
$\Rightarrow tan(2\times 22.5)=1$
if we assume $22.5=\theta$
$\Rightarrow tan2\theta =1$
we know $tan2\theta =\frac{2tan\theta }{1-ta{n}^2\theta }$
$\Rightarrow 1=\frac{2tan\theta }{1-ta{n}^2\theta }\Rightarrow 1-ta{n}^2\theta =2tan\theta \Rightarrow ta{n}^2\theta +2tan\theta -1=0\Rightarrow tan\theta =x{x}^2+2x-1=0x=\frac{-2\pm \sqrt{4+4}}{2}=\frac{-2\pm 2\sqrt{2}}{2}x=-1\pm \sqrt{2}\therefore tan22.5=-1+\sqrt{2}$