Question 1137387

Proof :

{{{tan (75) - tan( 60) = 2}}}

use: {{{tan (75)=tan(45+30)}}} and {{{tan( 60)=tan(30+30)}}}


{{{tan(45+30)-tan(30+30)=2}}}


recall: {{{tan(A + B)= (tan A + tan B) / (1 -tan A tan B)}}} 


{{{(tan(45) + tan (30)) / (1 -tan (45) tan(30)) -(tan(30) + tan (30)) / (1 -tan (30) tan(30)) =2}}}

since

{{{tan(45)=1}}} and

{{{tan(30)=1/sqrt(3)=sqrt(3)/3}}}

we have

{{{(1 +sqrt(3)/3) / (1 -1*sqrt(3)/3) -(sqrt(3)/3 + sqrt(3)/3) / (1 -(sqrt(3)/3)*(sqrt(3)/3)) =2}}}


{{{((3 +sqrt(3))/3) / ((3 -sqrt(3))/3) -(2sqrt(3)/3) / (1 -3/9) =2}}}


{{{(3 +sqrt(3))/ (3 -sqrt(3)) -(2sqrt(3)/3) / (3/3 -1/3) =2}}}


{{{(3 +sqrt(3))/ (3 -sqrt(3)) -(2sqrt(3)/3) / (2/3) =2}}}


{{{(3 +sqrt(3))/ (3 -sqrt(3)) -(2sqrt(3)) / 2 =2}}}


{{{(2(3 +sqrt(3)) -2sqrt(3)(3 -sqrt(3))) / 2(3 -sqrt(3))=2}}}


{{{(6 +2sqrt(3) -6sqrt(3) +2*3) / 2(3 -sqrt(3))=2}}}


{{{(12  -4sqrt(3) ) / 2(3 -sqrt(3))=2}}}


{{{4(3  -sqrt(3) ) / 2(3 -sqrt(3))=2}}}...simplify


{{{4/ 2=2}}}


{{{2=2}}}