To simplify this expression, we can use the double angle formula for cosine:

cos(2x) = 2cos^2(x) - 1

Now, let's substitute x = 45 - a/2:

cos(2(45 - a/2)) = 2cos^2(45 - a/2) - 1

cos(90 - a) = 2cos^2(45 - a/2) - 1

Since cos(90 - a) = sin(a), we have:

sin(a) = 2cos^2(45 - a/2) - 1

Now, we can simplify this expression further:

sin(a) = 2(cos^2(45)cos^2(a/2) - sin^2(45)sin^2(a/2)) - 1

sin(a) = 2((1/2)(cos(a/2))^2 - (1/2)(sin(a/2))^2) - 1

sin(a) = (cos(a/2))^2 - (sin(a/2))^2 - 1

sin(a) = cos(2a/2) - 1

sin(a) = cos(a) - 1

Therefore, the simplified expression is cos(a) - 1.