Using the trigonometric identity for the sum of angles, we have:

cos(a + b) = cos(a)cos(b) - sin(a)sin(b)

sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

Let's apply these identities to the expression cos(80)sin(10) + sin(80)cos(10):

cos(80)sin(10) + sin(80)cos(10)

= (cos(80)cos(80))sin(10) + (sin(80)sin(80))cos(10) (using the identity cos^2(a) + sin^2(a) = 1)

= cos(80)cos(10)sin(80)sin(10) + cos(80)cos(10)sin(80)sin(10)

= cos(80)cos(10)sin(80)sin(10) + cos(10)cos(80)sin(10)sin(80)

= 2cos(80)cos(10)sin(80)sin(10)

Now we can use the double angle identity for sine:

sin(2a) = 2sin(a)cos(a)

Let's substitute a = 10 degrees:

sin(20) = 2sin(10)cos(10)

sin(10)cos(10) = sin(20)/2

Substituting this back into the original expression:

2cos(80)cos(10)sin(80)sin(10)

= 2cos(80)sin(80)(sin(10)cos(10))

= 2cos(80)sin(80)(sin(20)/2)

= sin(160)/2

= sin(20)/2

Therefore, cos(80)sin(10) + sin(80)cos(10) = sin(20)/2.