We have to solve (tan x)^2 = 8 - 8*sec x.

(tan x)^2 = 8 - 8*sec x

=> (sin x)^2 / (cos x)^2 = 8 - 8/(cos x)

=> (1 - (cos x)^2) / (cos x)^2 = (8*(cos x)^2 - 8* cos x)/ (cos x)^2

=> (1 - (cos x)^2) = (8*(cos x)^2 - 8* cos x)

=> 9(cos x)^2 - 8 cos x - 1 = 0

=> 9(cos x)^2 - 9 cos x  + cos x - 1 = 0

=> 9(cos x)(cos x - 1) + 1( cos x - 1) = 0

=> (9(cos x) - 1)(cos x - 1) = 0

cos x = 1 and cos x = 1/9

x = arc cos 1 and x = arc cos (1/9) and - arc cos (1/9)

The required values are : x = 2*n*pi and x = arc cos (1/9) + 2*n*pi and x = -arc cos (1/9) + 2*n*pi