Question 151392
{{{tan(pi/6 + pi/4)}}} Start with the given expression.



{{{(tan(pi/6) + tan(pi/4))/(1-tan(pi/6)tan(pi/4))}}} Use the identity {{{tan(A + B)=(tan(A) + tan(B))/(1-tan(A)tan(B))}}} to expand the expression.




{{{(sqrt(3)/3 + 1)/(1-(sqrt(3)/3)(1))}}} Using the unit circle, evaluate {{{tan(pi/6)}}} to get {{{sqrt(3)/3}}} and evaluate {{{tan(pi/4)}}} to get {{{1}}}



{{{(sqrt(3)/3 + 1)/(1-sqrt(3)/3)}}} Multiply



{{{((sqrt(3)+3)/3)/((3-sqrt(3))/3)}}} Combine the fractions.



{{{((sqrt(3)+3)/3)(3/(3-sqrt(3)))}}}Multiply the first fraction by the reciprocal of the second fraction.



{{{((sqrt(3)+3)/cross(3))(cross(3)/(3-sqrt(3)))}}} Cancel out the common terms.



{{{(sqrt(3)+3)/(3-sqrt(3))}}} Simplify.



{{{((sqrt(3)+3)/(3-sqrt(3)))((3+sqrt(3))/(3+sqrt(3)))}}} Multiply the fraction by {{{(3+sqrt(3))/(3+sqrt(3))}}}



{{{((sqrt(3)+3)(3+sqrt(3)))/((3+sqrt(3))(3-sqrt(3)))}}} Combine the fractions.



{{{(3*sqrt(3)+3+9+3*sqrt(3))/(6)}}} FOIL



{{{(6*sqrt(3)+12)/(6)}}} Combine like terms



{{{(6(sqrt(3)+2))/(6)}}} Factor out the GCF {{{6}}}



{{{sqrt(3)+2}}} Reduce




So {{{tan(pi/6 + pi/4)=sqrt(3)+2}}}