Find the exact value:

4pi/16 = 1/root2

To find the exact value of the given expression, we can use the trigonometric identity for the cosine of a difference of angles:

cos(A - B) = cos(A)cos(B) + sin(A)sin(B)

Let's substitute A = pi/16 and B = 3pi/16 into this identity:

cos(pi/16 - 3pi/16) = cos(pi/16)cos(3pi/16) + sin(pi/16)sin(3pi/16)

Simplifying the left side:

cos(-2pi/16) = cos(-pi/8) = cos(pi/8)

Now we have:

cos(pi/8) = cos(45°)

To find the exact value of cos(45°), we can use the special triangle or unit circle.

Method 1: Using a Special Triangle
Draw a right-angled triangle with angles 45°, 45°, and 90°. The side lengths are in a ratio of 1:1:√2.

/|
/ |
/ |
/ _|_
/_
√2

Now we can see that the adjacent side length (the side adjacent to the angle of 45°) is 1, and the hypotenuse is √2.

By definition, cosine is equal to the adjacent side divided by the hypotenuse:

cos(45°) = 1/√2

Rationalizing the denominator, we multiply both the numerator and denominator by √2:

1/√2 * (√2/√2) = √2/2

So, cos(45°) = √2/2.

Method 2: Using the Unit Circle
On the unit circle, the angle of 45° corresponds to the point (cos(45°), sin(45°)) = (cos(π/4), sin(π/4)) = (1/√2, 1/√2).

Therefore, cos(45°) = 1/√2.

Now, substituting back this value into our original expression:

cos(pi/16)cos(3pi/16) - sin(pi/16)sin(3pi/16)
= (1/√2)(1/√2) - (1/√2)(1/√2)
= 1/2 - 1/2
= 0

Hence, the exact value of the given expression is 0.