1. Prove that cos 2x 2 cos2 x 1.
To prove that cos 2x = 2 cos^2x - 1, we can use the double angle formula for cosine:
cos 2x = cos^2x - sin^2x
Since sin^2x = 1 - cos^2x, we can substitute this expression into the formula for cos 2x:
cos 2x = cos^2x - (1 - cos^2x)
cos 2x = cos^2x - 1 + cos^2x
cos 2x = 2 cos^2x - 1
Therefore, we have proved that cos 2x = 2 cos^2x - 1.