The minimum value of $$\cos 2\theta + \cos \theta $$   for real values of $$\theta $$ is

Solution :
Value $$ = 2{\cos ^2}\theta - 1 + \cos \theta = - 1 + 2\left( {{{\cos }^2}\theta + \frac{1}{2}\cos \theta + \frac{1}{{16}}} \right) - \frac{1}{8}$$
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \frac{9}{8} + 2{\left( {\cos \theta + \frac{1}{4}} \right)^2} \geqslant - \frac{9}{8}.$$
So, the minimum value $$ = - \frac{9}{8}.$$