Use a sum or difference identity to find an exact value of sin 2pi over 12

5pi/12 = 3pi/12 + 2pi/12 = pi/4 + pi/6

sin (a + b) = > sin (a) cos (b) + sin (b) cos (a)

cos (a + b) = > cos (a) cos (b) - sin (a) sin (b)

sin (5pi/12) = >

sin (3pi/12 + 2pi/12) = >

sin (pi/4 + pi/6) = >

sin (pi/4) cos (pi/6) + sin (pi/6) cos (pi/4) = >

(sqrt (2) / 2) * (sqrt (3) / 2) + (1/2) * (sqrt (2) / 2) = >

(sqrt (2) / 4) * (1 + sqrt (3))

cos (5pi/12) = >

cos (pi/4 + pi/6) = >

cos (pi/4) cos (pi/6) - sin (pi/4) sin (pi/6) = >

(sqrt (2) / 2) * (sqrt (3) / 2) - (sqrt (2) / 2) * (1/2) = >

(sqrt (2) / 4) * (sqrt (3) - 1)

tan (5pi/12) = >

sin (5pi/12) / cos (5pi/12) = >

(sqrt (2) / 4) * (sqrt (3) + 1) / ((sqrt (2) / 4) * (sqrt (3) - 1)) = >

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

(sqrt (3) + 1) ^2 / (3 - 1) = >

(3 + 2 * sqrt (3) + 1) / 2 = >

(4 + 2 * sqrt (3)) / 2 = >

2 + sqrt (3)

csc (5pi/12) = >

1 / sin (5pi/12) = >

4 / (sqrt (2) * (sqrt (3) + 1)) = >

4 * sqrt (2) * (sqrt (3) - 1)) / (2 * (3 - 1)) = >

4 * sqrt (2) * (sqrt (3) - 1)) / (2 * 2) = >

sqrt (2) * (sqrt (3) - 1)

sec (5pi/12) = >

1/cos (5pi/12) = >

4 / (sqrt (2) * (sqrt (3) - 1)) = >

4 * sqrt (2) * (sqrt (3) + 1)) / (2 * (3 - 1)) = >

sqrt (2) * (sqrt (3) + 1)

cot (5pi/12) = >

1/tan (5pi/12) = >

1 / (2 + sqrt (3)) = >

(2 - sqrt (3)) / (4 - 3) = >

2 - sqrt (3)