How to prove that #tan112 1/2=-sqrt2-1#?

2 Answers
Narad T.
May 30, 2018

Please see the proof below

Explanation:

We need

#cos(2x)=2cos^2x-1=1-2sin^2x#

Therefore,

#cos(x/2)=sqrt((1+cosx)/2)#

#sin(x/2)=sqrt((1-cosx)/2)#

#tan(x/2)=sin(x/2)/cos(x/2)=sqrt((1-cosx)/(1+cosx))#

#=sqrt(((1-cosx)(1-cosx))/((1+cosx)(1-cosx)))#

#=sqrt((1-cosx)^2/(1-cos^2x))#

#=sqrt((1-cosx)^2/(sin^2x))#

And finally,

#tan(x/2)=(1-cosx)/sinx#

Here,

#x=225#

#cos(225)=-1/sqrt2#

#sin(225)=-1/sqrt2#

#tan(225/2)=(1-(-1/sqrt2))/(-1/sqrt2)=-(sqrt2+1)#

Jim G.
May 30, 2018

#"see explanation"#

Explanation:

#"using the "color(blue)"half angle identity"#

#•color(white)(x)tan(x/2)=+-sqrt((1-cosx)/(1+cosx))#

# 112 1/2" is in the second quadrant where"#

#tan(112 1/2)<0#

#tan(112 1/2)=-sqrt((1-cos225)/(1+cos225))#

#color(white)(xxxxxxxx)=-sqrt((1-(-cos45))/(1+(-cos45))#

#color(white)(xxxxxxxx)=-sqrt((1+1/sqrt2)/(1-1/sqrt2))#

#color(white)(xxxxxxxx)=-sqrt((sqrt2+1)/(sqrt2-1))#

#color(white)(xxxxxxxx)=-sqrt((sqrt2+1)^2/((sqrt2-1)(sqrt2+1))#

#color(white)(xxxxxxxx)=-sqrt((sqrt2+1)^2)#

#color(white)(xxxxxxxx)=-(sqrt2+1)=-sqrt2-1#

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