CosA+cosB-cosC=4COSA/2 cosB/2 SINC/2-1

To simplify the given expression, we can first use the compound angle formula for cosine, which states that cos(A + B) = cosA*cosB - sinA*sinB. Similarly, cos(A - B) = cosA*cosB + sinA*sinB.

Let's rewrite the given expression using these formulas:

cosA + cosB - cosC = 4cos(A/2)cos(B/2)sin(C/2) - 1

Now, let's multiply both sides of the equation by 2:

2cosA + 2cosB - 2cosC = 8cos(A/2)cos(B/2)sin(C/2) - 2

Next, let's add 2cosC to both sides:

2cosA + 2cosB = 8cos(A/2)cos(B/2)sin(C/2) + 2cosC - 2

Now, let's factor out 2 from the right side:

2cosA + 2cosB = 2(4cos(A/2)cos(B/2)sin(C/2) + cosC - 1)

Finally, we can cancel out the 2 on both sides:

cosA + cosB = 4cos(A/2)cos(B/2)sin(C/2) + cosC - 1

Therefore, the simplified expression is:

cosA + cosB = 4cos(A/2)cos(B/2)sin(C/2) + cosC - 1.