The half angle formula for cosine states that cos(x/2) = ±sqrt((1 + cos(x))/2), where the sign depends on the quadrant of the angle x/2.

To find the exact value of cos 30 degrees using the half angle formula, we can start by using the identity cos(2x) = 1 – 2sin^2(x) or cos(2x) = 2cos^2(x) – 1.

Let x = 15 degrees. Then we have:

cos(30) = cos(2x) = 2cos^2(x) – 1 = 2cos^2(15) – 1

Now, we can use the half angle formula for cosine with x = 15 degrees:

cos(15) = ±sqrt((1 + cos(30))/2)

Since cos(30) is positive in the first quadrant, we take the positive root:

cos(15) = sqrt((1 + cos(30))/2)

Squaring both sides, we get:

cos^2(15) = (1 + cos(30))/2

Substituting this expression into our earlier equation for cos(30), we get:

cos(30) = 2cos^2(15) – 1 = 2(1 + cos(30))/2 – 1 = cos(30) + 1 – 1 = cos(30)

Therefore, we have shown that cos(30) = cos(30), which is a tautology.

So, the exact value of cos 30 degrees is 1/2.