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

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Let x=cos⁡2θ+cos⁡θx=2cos2⁡θ+cos⁡θ−1⇒x=−1+2cos2⁡θ+12cos⁡θ⇒x=−1+2cos⁡θ+142−116⇒x=−98+2cos⁡θ+142⇒x≥−98

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The minimum value of cos⁡2θ+cos⁡θ for real values of θ, is

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