[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)
