1. Prove that cos 2x 2 cos2 x 1.
2. Prove that cos 2x 1 2 sin2 x
To prove these identities, we will use the trigonometric identity for cosine and sine:
cos(2x) = cos^2(x) - sin^2(x) = 1 - 2sin^2(x) [1]
1. Proof of cos(2x) = 2cos^2(x) - 1:
Start with the right side of the equation:
2cos^2(x) - 1
Using the double angle formula for cosine:
2cos^2(x) - 1 = 2(cos^2(x) - sin^2(x)) - 1
We know that cos(2x) = cos^2(x) - sin^2(x) from the trigonometric identity [1].
Substitute this into the equation:
2(cos^2(x) - sin^2(x)) - 1 = 2cos(2x) - 1
Therefore, 2cos^2(x) - 1 = cos(2x).
2. Proof of cos(2x) = 1 - 2sin^2(x):
Start with the right side of the equation:
1 - 2sin^2(x)
Using the double angle formula for cosine:
1 - 2sin^2(x) = cos^2(x) - 2sin^2(x)
Substitute cos^2(x) = 1 - sin^2(x) into the equation:
cos^2(x) - 2sin^2(x) = (1 - sin^2(x)) - 2sin^2(x)
Simplify the equation:
1 - sin^2(x) - 2sin^2(x) = 1 - 2sin^2(x)
Therefore, 1 - 2sin^2(x) = cos(2x).
Both identities have been proven using trigonometric identities and properties.