Cosx+2secx=-3
solve each equation for all real values of x

Cos x + 2sec x = -3
cos x + 2/cos x = -3
cos^2 x + 2 = -3cos x
cos^2 x + 3cos x + 2 = 0
(cos x + 1)(cos x + 2) = 0
cos x + 1 = 0 or cos x + 2 = 0
cos x = -1 or cos x = -2                      [but cos x cannot be -2, so we discard it]
x = arccos(-1)
x = (2n + 1)π