Find the exact value of tan 15 degrees

tan (15) =

sin (15) / cos (15) =

sin (45 - 30) / cos (45 - 30) =

[ sin (45) cos (30) - sin (30) cos (45) ] / [ cos (45) cos (30) + sin (45) sin (30) ]

Since sin (45) = cos (45) = √2/2, you can just factor that out from the top and bottom

[ cos (30) - sin (30) ] / [ cos (30) + sin (30) ]

[ √3/2 - 1/2 ] / [ √3/2 + 1/2]

(√3 - 1) / (√3 + 1)

(√3 - 1) ^2 / (√3+1) (√3 - 1)

(√3 - 1) ^2 / (3 - 1)

(3 - 2√3 + 1) / 2

2 - √3

There's also a formula for tan (a-b), but I couldn't remember it off hand.