Answer:
The exact value of [tex]tan(105\°)[/tex] is [tex]-2-\sqrt{3}[/tex]
Step-by-step explanation:
The half-angle identity for 'tangent' is.....
[tex]tan(\frac{\theta}{2})=\frac{sin(\theta)}{1+cos(\theta)}[/tex]
Here, [tex]\frac{\theta}{2}=105\° \Rightarrow \theta=210\°[/tex]
Plugging the value of [tex]\theta[/tex] into the above formula.....
[tex]tan(\frac{210\°}{2})=\frac{sin(210\°)}{1+cos(210\°)}\\ \\ tan(105\°)=\frac{-\frac{1}{2}}{1+(-\frac{\sqrt{3}}{2})}\\ \\ tan(105\°)=\frac{-\frac{1}{2}}{1-\frac{\sqrt{3}}{2}}\\ \\ tan(105\°)=\frac{-\frac{1}{2}}{\frac{2-\sqrt{3}}{2}}\\ \\tan(105\°)=-\frac{1}{2-\sqrt{3}}=-\frac{(2+\sqrt{3})}{(2-\sqrt{3})(2+\sqrt{3})}=-\frac{(2+\sqrt{3})}{4-3}=-\frac{(2+\sqrt{3})}{1}\\ \\ tan(105\°)=-2-\sqrt{3}[/tex]
