Find all solutions to the equation.

cos2x + 2 cos x + 1 = 0

For cos (2x) * (2cos (x) + 1) = 0, use the double angle identity for cos (2x), which is cos^2 x - sin^2 x = cos^2 x - (1-cos^2) = 2cos^2 x - 1.

So we have (2cos^2 x - 1) (2cos x + 1) = 0. So 2cos^2 x - 1 = 0 or x = 0 and 2pi.

For 2sec^2 x + tan^2 x - 3 = 0, use the identity sec^2 x = tan^2 x + 1, so we have

2 (tan^2 x + 1) + tan^2 x - 3 = 0 or

2tan^2 x + tan^2 x - 1 = 0 or

3 tan^2 x = 1.

So x = pi/2, pi/2 + pi = 3pi/2.