How to prove that tan112 1/2=-sqrt2-1?
2 Answers
Please see the proof below
Explanation:
We need
Therefore,
And finally,
Here,
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