Find the exact value of tan 15 and cos 105 without using tables or a calculator.I honestly don't know how to work this out. I can't find an example...

We could calculate tan 15 = sin 15/ cos
15.

sin 15/ cos 15 = sin (45-30) / cos
(45-30)

We can calculate sin 15 = sin
(45-30)

 = sin 45*cos 30 - sin 30*cos
45

sin 45 = sqrt2/2 = cos
45

sin 30 = 1/2

cos 30 =
sqrt3/2

sin (45-30) = sqrt6/4 - sqrt2/4 = sqrt2(sqrt3 -
1)/4

cos (45-30) = cos 45cos30 +
sin45sin30

cos (45-30) = sqrt6/4 +
sqrt2/4

cos (45-30) =
sqrt2(sqrt3+1)/4

tan 15 = [sqrt2(sqrt3 -
1)/4]*[4/sqrt2(sqrt3+1)]

tan 15 = (sqrt3 - 1)/(sqrt3 +
1)

tan 15 = (sqrt3 - 1)^2/(3 -
1)

tan 15 =
(4-2sqrt3)/2

tan 15 =
2-sqrt3

We'll calculate cos 105 = cos
(15+90)

 cos (15+90) = cos15*cos90 -
sin15*sin90

cos 90 = 0

sin 90
= 1

sin 15 = sqrt2(sqrt3 -
1)/4

cos (15+90) = - sqrt2(sqrt3 -
1)/4

cos 105 = sqrt2(
1-sqrt3)/4