Question

4cosasina+cos^2(a)=-2

Answer 1

Sorry - this function has a minimum value of -1.56

So, no solutions.

But, you can fix it, and follow the steps below to find a solution.
4sina cosa + cos^2a = -2
2sin2a + (cos2a-1)/2 = -2
4sin2a + cos2a - 1 = -4
4sin2a + cos2a = -3
√17 (sin2a * 4/√17 + cos2a * 1/√17) = -2
√17 sin(2a + arcsin 1/√17) = -2
sin(2a + arcsin 1/√17) = -2/√17
2a + arcsin 1/√17 = -arcsin 2/√17
a = -(arcsin 1/√17 + arcsin 2/√17)