Answer:
2 - sqrt(3)
Step-by-step explanation:
Split pi/12 into two angles where the values of the six trigonometric functions are known.
tan (pi/4 - pi/6)
Apply the difference of angles identity
[tex]\frac{tan(pi/4) - tan(pi/6)}{1 + tan(pi/4)tan(pi/6)}[/tex]
tan(pi/4) = 1 , tan(pi/6) = (sqroot3)/3
Plug in and Simplify
[tex]\frac{1-\frac{\sqrt{3} }{3} }{1+1\frac{\sqrt{3} }{3} }[/tex]
[tex]\frac{\frac{3-\sqrt{3} }{3} }{\frac{3+\sqrt{3} }{3} }[/tex]
[tex]\frac{3-\sqrt{3} }{3} *\frac{3}{3+\sqrt{3} }[/tex]
[tex]\frac{3-\sqrt{3} }{3+\sqrt{3}}[/tex] Need to multiply this by [tex]\frac{3+\sqrt{3} }{3+\sqrt{3} }[/tex]
Expand and simplify numerator: [tex]\frac{6}{(3+\sqrt{3} )^{2} }[/tex]
Expand and simplify denominator: [tex]\frac{6}{12+6\sqrt{3}}[/tex]
Cancel the common factor: [tex]\frac{1}{2+\sqrt{3}}[/tex]
