There exists a value of $$\theta $$ between 0 and $$2\pi $$ that satisfies the equation $$\,\,{\sin ^4}\theta - 2{\sin ^2}\theta - 1 = 0.$$

If the complex numbers, $${Z_1},{Z_2}$$ and $${Z_3}$$ represent the vertics of an equilateral triangle such that
$$\left| {{Z_1}} \right| = \left| {{Z_2}} \right| = \left| {{Z_3}} \right|$$ then $${Z_1} + {Z_2} + {Z_3} = 0.$$

Find the values of $$x \in \left( { - \pi , + \pi } \right)$$ which satisfy the equation $${g^{(1 + \left| {\cos x} \right| + \left| {{{\cos }^2}x} \right| + \left| {{{\cos }^3}x} \right| + ...)}} = {4^3}$$

If a < b < c < d, then the roots of the equation (x - a) (x - c) + 2 ( x - b) (x - d) = 0 are real and distinct.