What are all the exact solutions of 2sec^2x-tan^4x=-1 ?

The first thing to do would be to simplify the equation first. Let's apply the trigonometric functions properties where 1 + tan²x = sec² x. So, we can substitute this to the equation so that it would purely be a function of tangents.

2sec²x-tan⁴x=-1
2(1+tan²x) - tan⁴x = 1
2 + 2 tan²x - tan⁴x + 1 = 0
tan⁴x - 2tan²x + 1 = 0
(tanx -1)² = 0
tanx - 1 = 0
tanx = 1
x = tan⁻¹ 1
x = 45°

So, the solution for the given trigonometric equation is 45° or π/4 radians.