• question_answer $\cos \,\,2\theta +2\,\,\cos \theta$ is always A) Greater than $-\frac{3}{2}$ B) Less than or equal to $\frac{3}{2}$ C) Greater than or equal to $-\frac{3}{2}$ and less than or equal to 3 D) None of these

Solution :

We have $\cos 2\theta +2\cos \theta =2{{\cos }^{2}}\theta -1+2\cos \theta$             $=2{{\left( \cos \theta +\frac{1}{2} \right)}^{2}}-\frac{3}{2}$ Now $2{{\left( \cos \theta +\frac{1}{2} \right)}^{2}}\ge 0$for all $\theta$ $\therefore \,\,2{{\left( \cos \theta +\frac{1}{2} \right)}^{2}}-\frac{3}{2}\ge \frac{-3}{2}$for all $\theta$. Þ $\cos 2\theta +2\cos \theta \ge \frac{-3}{2}$for all $\theta$

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