Question

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

• A. $-\frac{9}{8}$
• B. 0
• C. $-2$
• D. none of these.

Answer 1

Answer: (A)

$-\frac{9}{8}$

$cos2\theta +cos\theta =2\ast co{s}^2\theta -1+cos\theta$
$=2[cos\theta +\frac{cos\theta }{2}+\frac{1}{16}-\frac{1}{16}]-1$
$=(2cos\theta +\frac{1}{4}{)}^2-\frac{1}{8}-1$
$=(2cos\theta +\frac{1}{4}{)}^2-\frac{9}{8}$
For minimum value $cos\theta =-\frac{1}{4}$
Therefore, minimum value $=-\frac{9}{8}$

Hence, option 'A' is correct.