Calculate tan75 in two methods.

One way you can find tan 75 is by using the relation     
                 tan(a+b) = {tan(a)
+tan(b)}/{1-tan(a)tan(b)}.

Take a as 45 and b as
30.

tan 75= tan(45+30) = {tan(30)
+tan(45)}/{1-tan(45)(tan(30)}

Now tan 45 is 1 and tan 30 is
1/3^(1/2)

Substituting these we get {1/3^(1/2)
+1}/{1-1/3^(1/2)}

Or tan 75=
{1+1/3^(1/2)}/{1-1/3^(1/2)}

Another
method would be using

tan 75=sin 75/cos
75=sin(45+30)/cos(45+30)=
(sin45cos30+cos45sin30)/(cos45cos30-sin45sin30)

Now,
cos45=sin45, so we can eliminate these terms and we get
(cos30+sin30)/(cos30-sin30)

Now cos 30=3^(1/2)/2 and sin
30=1/2

So
substituting:

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

tan
75= {3^(1/2)+1}/{3^(1/2)-1}