To find the value of

cos(11π/12) + cos(5π/12),

we can use the following trigonometric identity:

cos(a) + cos(b) = 2 cos((a + b)/2) cos((a - b)/2).

So,

cos(11π/12) + cos(5π/12)

= 2 cos((11π/12 + 5π/12)/2) cos((11π/12 - 5π/12)/2)

= 2 cos(8π/12) cos(3π/12)

= 2 cos(2π/3) cos(π/4)

= 2 (-1/2) (√2/2)

= -√2.

Therefore, the value of cos(11π/12) + cos(5π/12) is -√2.