Answer:
C
Step-by-step explanation:
Using the addition formula for tangent
tan(x + y) = [tex]\frac{tanx+tany}{1-tanxtany}[/tex] and the exact values
tan([tex]\frac{\pi }{6}[/tex] ) = [tex]\frac{1}{\sqrt{3} }[/tex] and tan([tex]\frac{\pi }{4}[/tex] ) = 1
tan([tex]\frac{5\pi }{12}[/tex] ) = tan([tex]\frac{\pi }{6}[/tex] + [tex]\frac{\pi }{4}[/tex] )
= [tex]\frac{tan(\frac{\pi }{6})+tan(\frac{\pi }{4}) }{1-tan(\frac{\pi }{6})tan(\frac{\pi }{4}) }[/tex]
= [tex]\frac{\frac{1}{\sqrt{3} }+1 }{1-\frac{1}{\sqrt{3} }.1 }[/tex] ( multiply numerator/ denominator by [tex]\sqrt{3}[/tex] )
= [tex]\frac{1+\sqrt{3} }{\sqrt{3} -1}[/tex]
rationalise denominator by multiplying numerator/denominator by ([tex]\sqrt{3}[/tex] + 1)
= [tex]\frac{(1+\sqrt{3})(\sqrt{3}+1) }{(\sqrt{3}-1)(\sqrt{3}+1) }[/tex]
= [tex]\frac{\sqrt{3}+1+3+\sqrt{3} }{3-1}[/tex]
= [tex]\frac{2\sqrt{3}+4 }{2}[/tex]
= [tex]\sqrt{3}[/tex] + 2 → C
