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 ......and this is the exact value......
