The trigonometric function is given as
[tex]\tan 75^{\circ}[/tex]Apply the half angle identity to find the value of tan 75 ,
[tex]\tan (\frac{u}{2})=\frac{\sin u}{1+\cos u}[/tex]Here,
[tex]\tan (\frac{150^{\circ}}{2})=\frac{\sin150^{\circ}}{1+cos150^{\circ}}[/tex][tex]\tan (75^{\circ})=\frac{\frac{1}{2}}{1-\frac{\sqrt[]{3}}{2}}=\frac{\frac{1}{2}}{\frac{2-\sqrt[]{3}}{2}}^{}[/tex][tex]\tan 75^{\circ}=\frac{1}{2-\sqrt[]{3}}[/tex]Now rationalize the function.
[tex]\tan 75^{\circ}=\frac{1}{2-\sqrt[]{3}}\times\frac{2+\sqrt[]{3}}{2+\sqrt[]{3}}=\frac{2+\sqrt[]{3}}{4-3}=\frac{2+\sqrt[]{3}}{1}[/tex]Again simplify the trigonometric function,
[tex]\tan 75^{\circ}=2+1.732=3.732[/tex]Hence the answer is 3.732.
