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AonZ
Given t = tan112.5, show that 2t / 1-t^2 = 1 hence show that tan112.5 = -sqrt(2) - 1
use tan trig identity itll simpify
\[\tan 2t=\frac{2\tan t }{1-\tan ^{2} t },put t=112.5\] tan2t=tan(112.5)*2=tan225=tan(180+45)=tan45=1 \[\frac{ 2\tan t }{1-\tan ^{2}t }=\1\] \[\frac{ 2t }{1-t ^{2} }=1\] 2t=1-t^2 \[t ^{2}+2t-1=0,t=\frac{ -2\pm \sqrt{2^{2}-4\left( 1 \right)\left( -1 \right)} }{2\times1}\] \[t=\frac{ -2\pm \sqrt{8} }{2 }=\frac{ -1\pm \sqrt{2} }{ 1 }\] in the second quadrant tan is negative. \[Hence \tan112.5=-1-\sqrt{2}\]