exact value

Question

use an angle sum identity to calculate the exact value of  
tan(75).

Answer 1

tanβ = sinβ / cosβ ,
sin45⁰ = cos45⁰ = (√2)/2 ---> tan45⁰ = 1
sin30⁰ = 1/2 , cos30⁰ = √3 / 2 ---> tan30⁰ = 1/√3

tan(45⁰ + 30⁰) = (tan45 + tan30) / (1 - tan45·tan30) = (1 + 1/√3) / (1 - 1/√3) =
(√3 + 1) · (√3 + 1)        (√3 + 1)²         (√3)² + 2√3 + 1 
 (√3 - 1) · (√3 + 1)        (√3)² - 1²                  2   

  2(2 + √3)                2 + √3   <----- is the final answer
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