Given
sec ² (x)-2=tan ² (x)
we look for solutions x ∈ [0,2 π ]
First, rewrite equation in sin(x) and cos(x),
1/cos²(x) - 2 = sin²(x)/cos²(x)
Multiply both sides by cos²(x), when cos(x)≠0, i.e. x≠ π/2 or 3π/2.
1-2cos²(x) = sin²(x)
Rearrange and solve:
1=(sin²(x)+cos²(x))+cos²(x)
=>
cos²(x)=0
=>
cos(x)=0
Since it is a condition before multiplication that cos(x)≠0, we conclude that there is no solution.
