Use a halfangle formula to find the exact value of csc 5pi12

Use a half-angle formula to find the exact value of csc 5pi/12?

Solution

csc(5pi/12)

Half angle formula : sin(x/2) = sqrt((1-cosx)/2)

where x= 2(5pi/12) = 5pi/6 =pi-pi/6

cos(5pi/6) = cos(pi-pi/6) = -cospi/6 = -sqrt3/2

sin(5pi/12) = sqrt(( 1 +sqrt3/2))/2

= sqrt((2 + sqrt3))/2