#[1]" "=csc(pi/12)#
Reciprocal Identity: #csctheta=1/sintheta#
#[2]" "=1/sin(pi/12)#
Represent #pi/12# as a difference of two special angles.
#[3]" "=1/sin((3pi)/12-(2pi)/12)#
#[4]" "=1/sin(pi/4-pi/6)#
Difference Identity: #sin(alpha-beta)=sinalphacosbeta-cosalphasinbeta#
#[5]" "=1/(sin(pi/4)cos(pi/6)-cos(pi/4)sin(pi/6))#
You can solve these since #pi/4# and #pi/6# are special angles.
#[6]" "=1/((sqrt2/2)(sqrt3/2)-(sqrt2/2)(1/2))#
#[7]" "=1/((sqrt6/4)-(sqrt2/4))#
#[8]" "=1/((sqrt6-sqrt2)/4)*4/4#
#[9]" "=4/(sqrt6-sqrt2)#
Rationalize the denominator.
#[10]" "=4/(sqrt6-sqrt2)*(sqrt6+sqrt2)/(sqrt6+sqrt2)#
#[11]" "=(4(sqrt6+sqrt2))/(6-2)#
#[12]" "=(cancel4(sqrt6+sqrt2))/cancel(4)#
#[13]" "=color(blue)(sqrt6+sqrt2)#