cos22.5=1/2 sqrt(A + sqrt B)

To find the value of cos(22.5), we need to use the trigonometric identity known as the Half-Angle formula for cosine.

The Half-Angle formula for cosine states that cos(x/2) = ±sqrt((1 + cos(x))/2).

Let's substitute x = 45 to find cos(22.5):

cos(22.5) = cos(45/2)

Now, we know that cos(45) = sqrt(2)/2, so we can substitute it into the formula:

cos(22.5) = cos(45/2) = ±sqrt((1 + cos(45))/2)

cos(45) = sqrt(2)/2

cos(22.5) = ±sqrt((1 + sqrt(2)/2)/2)

Next, we need to simplify the expression inside the square root:

A = 1 + sqrt(2)/2

B = 2

cos(22.5) = ±sqrt(A + sqrt(B)/2)

Therefore, cos(22.5) = ±1/2 * sqrt(A + sqrt(B)), where A = 1 + sqrt(2)/2 and B = 2.