Tan Tan

How do you solve for tan 75 when using sum and difference formula?

I know that tan 75= tan (45+30)=tan (A+B)
so, (tan 45+tan 30) / (1-tan 45 tan 30) =
(1+1/radical 3) / (1-1 times 1/radical 3) then from there my math gets all messed up. Can anyone show me the correct way to finish this and end up with the answer: 2+radical 3? Thank you!!!!!!!!!!!!!

tan(75)
= tan(45 + 30)
=(tan 45 + tan 30) / (1- tan 45 tan 30)
= (1 + 1/sqrt(3)) / (1 - 1/sqrt(3))

You can rationalize the denominators of both of those which gives you:

(1 + sqrt(3)/3) / (1 - sqrt(3)/(3))

Finding common denominators gives you:

[(3 + sqrt(3)) / 3] / [(3 - sqrt(3) / 3].

You can use the "keep change flip" rule for complex fractions to get:

[(3 + sqrt(3)) / 3] * [3 / (3 - sqrt(3))].

The 3's cancel and leave you with:

[3 + sqrt(3)] / [3 - sqrt(3)]

We can rationalize this again by multiplying top and bottom by 3+sqrt(3) which leaves us:

(3 + sqrt(3))^2 / [(3 + sqrt(3))(3 - sqrt(3))]
= (9 + 6sqrt(3) + 3) / [9 - 3]
= (12 + 6sqrt(3)) / (6)
2 + sqrt(3)

Does that make sense?

And note: my sqrt(3) is the same as your radical(3).

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