sec = 1/cos
#cos ((-11pi)/12) = cos ((11pi)/12) = cos ((3pi)/12 + (8pi)/12)# =
#= cos (pi/4 + (2pi)/3)#
Use the trig identity: cos (a + b) = cos a.cos b - sin a.sin b
#cos a = cos (pi/4) = sqrt2/2# ; #cos b = cos ((2pi)/3) = -1/2#
#sin a = sin (pi/4) = sqrt2/2# ; #sin b = sin ((2pi)/3) = sqrt3/2#
Therefor:
#cos ((-11pi)/12) = cos (pi/4 + (2pi)/3) =#
# (sqrt2/2)(-1/2) - ((sqrt2)/2)((sqrt3)/2)# = #- (sqrt2 + sqrt6)/4#
Check by calculator:
#cos ((-11pi)/12) = cos (-165) = - 0.966#
#- (sqrt2 + sqrt6)/4 = - 0.966.# OK
Finally:# sec ((-11pi)/12) = - 4/(sqrt2 + sqrt6)# = #- 1/0.966 = - 1.035#
