Determine the value of cot15 and sin75.

We know that cotx =
cosx/sinx

==> cot 15 = cos 15 /
sin15

Let us
rewrite:

==> cot15 =
(cos(45-30)/sin(45-3)

We know
that:

cos(a-b) = cosa8cosb +
sina*sinb

==> cos(45-30) = cos45*cos30 +
sin45*sin30

                           = sqrt(2)/2 *
sqrt3/2 + sqrt2/2 * 1/2

                             =
sqrt6/4 + sqrt2/4

                               =
(sqrt6+sqrt2)/4

Also we know
that;

sin(a-b) = sina*cosb -
sinb*cosa

sin(45-30) = sin45*cos30 -
sin30*cos45

                   = sqrt2/2 * sqrt3/2 -
1/2*sqrt2/2

                   = sqrt6/4 -
sqrt2/4

                    =
(sqrt6-sqrt2)/4

Now we will subsitute
:

cot15 = (sqrt6+sqrt2)/4 /
(sqrt6-sqrt2)/4

           =
(sqrt6+sqrt2)/(sqrt6-sqrt2)

            = (sqrt6 +sqrt2)^2
/ (6-2)

               = 6 + 2sqrt12 + 2 )/
4

              = (8+
4sqrt3)/4

                = 2+
sqrt3

==> cot 15 = 2 +
sqrt3

sin75 = sin(90 -
15)

sin(a-b) =  sina*cosb -
sinb*cosa

sin(90-15) = sin90*cos15 -
sin15*cos90

                 = 1*cos15 -
sin15*0

                    =
cos15

==> sin75 = cos15 =
(sqrt6+sqrt2)/4