sec^2(x) + tan(x) = 3 => tan^2(x) + 1 + tan(x) = 3 => tan^2(x) + tan(x) - 2 = 0 => [tan(x) + 2][tan(x) - 1] = 0 => plenty of real solutions for x.

sin^2(x) + 3cos^2(x) = 0: Restriciting to real numbers, the sum of two squares is always greater than or equal to 0. But sin^2(x) and cos^2(x) cannot simultaneously equal 0 for any real value of x. Therefore .....