when solving for all solutions how do I work out this problem: cos4(alpha)=cos2(alpha)

Question

and when verifying each identity how do I complete this problem: sin (alpha/2)=sec alpha -1/2sin (alpha/2)sec alpha

Answer 1

cos4x = cos2x
2cos^2(2x)-1 = cos2x
2cos^2(2x)-cos(2x)-1 = 1
(2cos2x+1)(cos2x-1) = 0
so, cos2x = -1/2 or 1
cos 2pi/3 = cos 4pi/3 = -1/2
cos 0 = 1

as for the identity, I get the feeling there's a typo, because secx appears twice on the right.