To simplify the expression cos3a + cos4a + cos5a, we can use the trigonometric identity cos(A+B) = cosAcosB - sinAsinB.

We can rewrite the expression as:

cos3a + cos4a + cos5a
= cos(3a) + cos(4a) + cos(5a)
= (cos(a + 2a)) + (cos(2a + 2a)) + (cos(2a + 3a))
= (cos(a)cos(2a) - sin(a)sin(2a)) + (cos(2a)cos(2a) - sin(2a)sin(2a)) + (cos(2a)cos(3a) - sin(2a)sin(3a))
= (cos(a)cos(2a) - sin(a)2sin(a)cos(a)) + (cos(2a)cos(2a) - sin(2a)2sin(2a)cos(2a)) + (cos(2a)cos(3a) - sin(2a)2sin(3a)cos(3a))

Now we can simplify this expression further by expanding and combining like terms.