√2cos^2(7π/12) - √2sin^2(7π/12)

Using the trigonometric identity cos^2(x) + sin^2(x) = 1, we can rewrite the expression as:

√2[cos^2(7π/12) - sin^2(7π/12)]

Now, we can use the double angle formula cos(2x) = cos^2(x) - sin^2(x) to simplify further:

√2cos(27π/12)

Simplifying the angle inside the cosine function:

√2*cos(7π/6)

Since the cosine function has period 2π, we can simplify the angle further:

√2*cos(π/6)

Using the value of cosine at π/6 (30 degrees) which is √3/2:

√2 * √3/2 = √6/2 = √6/2

Therefore, the final simplified expression is √6/2.