What is the exact value of tan 75°?

Step-by-step explanation:

Notice that tan (75) can be written as sin(75)/cos(75) = sin(30 + 45) / cos(30 + 45)

And using a couple of trig identities, we have

[sin 30 cos 45 + sin 45 cos 30 ] / [cos 30 cos 45 -sin 30 sin 45] =

[ (1/2)(1/√2) +(1/√2))(√3/2)] / [ (√3/2) (1/√2) -(1/2) (1/√2) ] =

([1 + √3)] / [2 √2]) / ([√3 - 1] / [2 √2]) =

[ 1 + √3] / [√3 - 1] rationalizing the denominator, we have

[ 1 + √3] * [√3 + 1] / 2 =

[ 1 + √3 ] [ 1 + √3 ] / 2 =

[1 + 2√3 + 3] / 2 =

[4 + 2√3 ] / 2 =

2 + √3 so this is the exact value