Remember that [tex] \sec^2t = \tan^2t + 1 [/tex]. This will help us in simplifying our problem.
Using the property above, we can simplify our expression by replacing the [tex] \sec^2t [/tex] with [tex] \tan^2t + 1 [/tex]:
[tex] (\tan^2t + 1) + \sqrt{3}\tan t - 1 = 0 [/tex]
[tex] \tan^2t + \sqrt{3}\tan t = 0 [/tex]
Now, we can use our algebraic properties to solve the equation:
[tex] \tan t(\tan t + \sqrt{3}) = 0 [/tex]
Using the Zero Product Property, we can set both factors equal to 0:
[tex] \tan t = 0 [/tex]
[tex] \tan t + \sqrt{3} = 0 [/tex]
Now, we can solve both of our two equations individually:
[tex] \tan t = 0 [/tex]
[tex] t = 0^{\circ}, 180^{\circ}, 360^{\circ} [/tex]
[tex] \tan t + \sqrt{3} = 0 [/tex]
[tex] \tan t = -\sqrt{3} [/tex]
[tex] t = 120^{\circ}, 300^{\circ} [/tex]
The solutions for [tex] t [/tex] in the scope of the original problem are [tex] \boxed{t = 0^{\circ}, 120^{\circ}, 180^{\circ}, 300^{\circ}, 360^{\circ}} [/tex].
