√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.
√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.